#72265266

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁵
prime radical	C₂×#33531	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4517 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^10|a^2=h,b^2=g,c^2=h,f^2=i,g^2=j,i^2=j,b^a=bf,c^b=cghj,d^a=dgij,e^d=ej,f^a=fi,f^b=fi,f^c=fj,f^d=fj,g^a=gj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265322, #72307057
maximal normal subgroup of	#72265322, #72307057

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