#72256502

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#16600
commutator	C₂³×C₄	C₂³×C₄
Frattini, cosocle	C₂⁴×C₄	C₂⁴
prime radical	C₂⁴×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

5949 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)(7,11)(8,19,20,31)(9,14)(10,15)(12,18)(13,29,30,21)(16,32,33,35)(17,25)(22,28)(23,26)(24,36,27,34);(2,4)(5,7,9,17,10,18,23,22)(6,11,14,25,15,12,26,28)(8,16,21,34,20,33,29,36)(13,27,31,32,30,24,19,35);(2,4)(5,7)(6,12,15,11)(8,16)(9,22)(10,18)(13,24,30,27)(14,25,26,28)(17,23)(19,35,31,32)(20,33)(21,36)(29,34);(5,8)(6,13,15,30)(7,16)(9,21)(10,20)(11,24,12,27)(14,31,26,19)(17,34)(18,33)(22,36)(23,29)(25,35,28,32)

Polycyclic presentation:

2^10|a^2=f,b^2=g,c^2=i,d^2=i,g^2=j,b^a=be,c^a=cefh,c^b=cgi,d^a=dgj,d^b=di,d^c=di,e^d=ej,f^d=fj,g^c=gj,h^b=hj,h^c=hj,i^a=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256508, #72256536, #72256541
maximal normal subgroup of	#72256508, #72256536, #72256541

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