#72268445

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

5865 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=ij,c^2=ij,d^2=g,e^2=f,h^2=j,b^a=bh,c^b=cj,d^b=dj,d^c=dg,e^a=eij,e^b=ei,e^c=ef,e^d=efgi,f^d=fi,g^e=gi,h^a=hj,h^b=hj

Conjugacy classes & character table

97 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268455, #72268463, #72310832
maximal normal subgroup of	#72268455, #72268463, #72310832

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