#72236765

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁸ · Schur multiplier (H₂) = C₂²⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁸
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

At least 71105 conjugacy classes of subgroups; only the 417223 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 8 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=j,b^2=ij,c^2=j,g^2=j,h^2=ij,b^a=bi,c^b=cj,d^a=dj,d^c=dj,g^a=gij,g^b=gi,g^c=gij,g^d=gi,g^e=gj,g^f=gi,h^e=hij,h^f=hj

Conjugacy classes & character table

265 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72238707

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