#72249882

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23960
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂×#24285	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2929 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3515 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)(7,15,17,31)(8,20)(9,22)(11,23,24,34)(13,29)(14,28)(16,32)(18,19,21,33)(25,36)(26,27,30,35);(11,24)(23,34)(26,30)(27,35);(1,2)(3,4)(5,6)(7,11)(8,13)(9,14)(10,12)(15,23)(16,25)(17,24)(18,26)(19,27)(20,29)(21,30)(22,28)(31,34)(32,36)(33,35);(1,3,4,2)(5,7,16,18,9,19,32,31,10,17,8,21,22,33,20,15)(6,11,25,26,14,27,36,34,12,24,13,30,28,35,29,23)

Polycyclic presentation:

2^10|a^2=g,d^2=eghj,h^2=i,i^2=j,c^a=ce,c^b=cg,d^a=dehij,d^b=df,d^c=de,f^c=fj,g^d=gj,h^a=hi,h^d=hj,i^a=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249911, #72250436, #72252898
maximal normal subgroup of	#72249911, #72250436, #72252898

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