#72322253

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#30396
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#411841	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1866 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=f,d^2=i,g^2=j,h^2=i,i^2=j,b^a=bg,c^a=ch,c^b=cf,d^a=dgij,d^b=dej,d^c=dgij,e^c=ej,g^a=gj,g^b=gj,g^c=gj,g^d=gj,h^a=hi,h^b=hj,h^c=hij,i^a=ij,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322256, #72322676
maximal normal subgroup of	#72322256, #72322676

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