#72250756

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#23960
commutator, Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	#498867	C₂
LCS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #20438 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2319 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=egj,c^2=ghj,d^2=egij,e^2=i,f^2=j,g^2=j,h^2=j,i^2=j,b^a=bh,c^a=ceh,c^b=ceh,d^a=df,d^b=dfg,d^c=dehj,e^b=ej,e^c=ei,f^a=fj,f^b=fj,f^c=fj,g^a=gj,g^b=gj,g^c=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	#72250777, #72252712, #72252864
maximal normal subgroup of	#72250777, #72252712, #72252864

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