#72312804

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×#532	C₂³
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

2266 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)(5,6,8,15)(7,18,20,33)(9,24,23,16)(10,17,11,12)(13,25,30,21)(14,26,29,27)(19,34,22,32)(28,35,31,36);(6,12,15,17)(13,28,30,31)(14,24,29,16)(18,34,33,32);(1,2,4,3)(5,7,8,20)(6,13,15,30)(9,25,23,21)(10,19,11,22)(12,28,17,31)(14,32,29,34)(16,33,24,18)(26,36,27,35);(1,3,4,2)(6,14,15,29)(7,19)(10,11)(12,24,17,16)(13,18,30,33)(20,22)(21,35)(25,36)(26,27)(28,34,31,32)

Polycyclic presentation:

2^10|a^2=e,b^2=i,c^2=e,d^2=eij,f^2=j,g^2=j,h^2=j,b^a=bh,c^a=cf,d^a=dfgj,d^c=dgi,f^a=fj,f^b=fi,f^c=fj,g^a=gj,g^c=gj,g^d=gj,h^a=hj,h^c=hij,h^d=hij,h^f=hj,i^a=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312811, #72312839, #72312896
maximal normal subgroup of	#72312811, #72312839, #72312896

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, 2, F.9);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.5*F.9*F.10);;
gap> SetPower(coll, 6, F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.7*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.9);;
gap> SetConjugate(coll, 6, 1, F.6*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, 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.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 4, F.8*F.9*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);