#72268359

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂⁴×C₄
Frattini, cosocle	C₂³×C₄	C₂⁵
prime radical	C₂×#61454	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

6000 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)(11,16,14,15)(17,18,22,25)(19,26,29,31)(20,24,21,30)(23,27,32,28);(5,13)(8,10)(11,14)(15,16)(17,19)(18,23)(20,28)(21,27)(22,29)(24,31)(25,32)(26,30);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,16,14,15)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,32,31);(1,4)(2,7)(3,9)(5,14)(6,12)(8,16)(10,15)(11,13)(17,20,22,21)(18,24,25,30)(19,28,29,27)(23,31,32,26);(1,5)(2,8)(3,10)(4,11,12,14)(6,13)(7,15,9,16)(18,25)(23,32)(24,30)(26,31)

Polycyclic presentation:

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

Conjugacy classes & character table

121 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72268377, #72268463, #72310565
maximal normal subgroup of	#72268377, #72268463, #72310565

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