#72312467

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#24260
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

2346 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)(6,13)(7,16)(8,19)(9,25)(10,11)(14,29)(15,18)(20,33)(22,31)(23,24)(26,30)(34,36);(5,6)(7,10)(8,20)(9,15)(11,13)(12,16)(14,22)(17,31)(18,27)(19,32)(21,34)(23,35)(24,33)(25,30)(26,28)(29,36);(5,7,11,16,12,13,10,6)(8,21,23,14,24,17,19,29)(9,26,28,15,25,18,27,30)(20,22,32,34,33,36,35,31);(1,3,4,2)(5,8,12,24)(6,14,16,29)(7,17,13,21)(9,22,25,36)(10,19,11,23)(15,20,30,33)(18,32,26,35)(27,31,28,34)

Polycyclic presentation:

2^10|a^2=e,c^2=gi,d^2=ej,g^2=j,h^2=j,b^a=bg,c^a=cgi,c^b=cg,d^a=dh,d^b=df,d^c=dj,f^a=fi,f^c=fj,g^a=gj,g^b=gj,g^d=gj,h^a=hj,h^b=hi,h^d=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312504, #72312668, #72312768
maximal normal subgroup of	#72312504, #72312668, #72312768

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