#72257415

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#32468
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂⁵	#137
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1374 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^10|b^2=fhj,c^2=hi,d^2=egj,e^2=i,h^2=j,i^2=j,b^a=bf,c^b=cfg,d^a=dfghi,d^b=dgij,d^c=de,e^c=ei,f^d=fj,g^b=gj,h^c=hj,h^d=hj,i^c=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257420, #72257434, #72257680
maximal normal subgroup of	#72257420, #72257434, #72257680

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