#72321808

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, socle	C₂²	#32468
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#525	D₄
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1388 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:

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

Polycyclic presentation:

2^10|a^2=h,b^2=e,d^2=gi,e^2=j,h^2=i,i^2=j,b^a=bhij,c^a=cg,c^b=cf,d^a=dh,d^b=dg,d^c=defi,e^a=ej,f^d=fj,g^d=gj,h^b=hi,h^d=hij,i^b=ij,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321841, #72322003, #72322844
maximal normal subgroup of	#72321841, #72322003, #72322844

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