#72321978

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#3184
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂²	#23761
prime radical	C₂×#3050	C₂²
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#1752 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1614 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=efgi,d^2=gij,e^2=ij,f^2=ij,g^2=j,h^2=ij,i^2=j,b^a=bf,c^a=cehi,c^b=cfgh,d^a=dei,d^b=dfghj,d^c=dgij,e^b=ei,e^d=eij,f^b=fij,f^d=fi,g^a=gj,g^b=gj,g^d=gj,h^b=hij,h^d=hi,i^b=ij,i^d=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322003, #72322428
maximal normal subgroup of	#72322003, #72322428

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