#72312475

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#25771
commutator	C₂×#41	C₂³×C₄
Frattini, cosocle	C₂²×#41	C₂⁴
prime radical	C₂×#3058	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

2336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6,9,15)(7,18,20,32)(8,24,25,28)(10,13,12,17)(11,16,26,29)(14,19,31,35)(21,33,23,30)(22,27,36,34);(6,13)(7,19)(8,11)(14,30)(15,17)(16,29)(18,34)(20,35)(21,36)(22,23)(24,28)(25,26)(27,32)(31,33);(1,2,4,3)(5,7,12,23)(6,14,17,34)(8,22,11,19)(9,21,10,20)(13,27,15,31)(16,33,24,32)(18,29,30,28)(25,35,26,36);(1,3,4,2)(5,8)(6,13,17,15)(9,25)(10,26)(11,12)(14,30,34,18)(16,29,24,28)(19,22)(27,33,31,32)(35,36)

Polycyclic presentation:

2^10|a^2=e,c^2=ef,d^2=efi,g^2=j,h^2=j,b^a=bf,c^a=cfhj,c^b=cg,d^a=dfhij,d^b=dj,d^c=dgj,e^c=ei,e^d=eij,f^c=fi,f^d=fij,g^a=gj,g^b=gj,g^c=gj,g^d=gj,h^a=hi,h^b=hi,h^c=hi,h^d=hi,h^e=hj,h^f=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312504, #72312627, #72312839
maximal normal subgroup of	#72312504, #72312627, #72312839

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.5);;
gap> SetPower(coll, 3, F.5*F.6);;
gap> SetPower(coll, 4, F.5*F.6*F.9);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6*F.8*F.10);;
gap> SetConjugate(coll, 3, 2, F.3*F.7);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.9*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 2, F.8*F.9);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);