#72265036

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂³×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂²×#447	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#399 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|a^2=i,b^2=f,d^2=fg,e^2=hj,i^2=j,b^a=bij,c^a=ci,d^a=dh,d^b=dfg,e^c=eh,e^d=ef,i^b=ij,i^c=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265187
maximal normal subgroup of	#72265187

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