#72312165

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁵
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

5756 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)(3,10)(4,14)(5,8)(6,9)(7,13)(11,15)(12,16)(17,18)(19,22)(20,24)(21,25)(23,26)(27,32)(28,31)(29,30);(1,3,6,13)(2,7,9,10)(4,12,5,11)(8,16,14,15)(17,19,22,18)(20,24,32,27)(21,25,31,28)(23,26,30,29);(17,20)(18,24)(19,27)(21,30)(22,32)(23,31)(25,26)(28,29);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(17,21)(18,25)(19,28)(20,30,32,23)(22,31)(24,26,27,29);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(18,19)(20,31,32,21)(23,30)(24,28,27,25)

Polycyclic presentation:

2^10|b^2=f,d^2=fhij,e^2=fij,b^a=bf,c^a=ch,c^b=cj,d^a=dfhij,d^b=df,d^c=dhj,e^a=eij,e^b=efi,e^c=eg,e^d=egj,g^a=gi,g^b=gj,g^d=gi,g^e=gj,h^d=hj,h^e=hi,h^g=hj,i^c=ij,i^d=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312180, #72313119
maximal normal subgroup of	#72312180, #72313119

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