#72339848

order	1024
nilpotency class	6
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#13561
commutator	C₄⋊Q₈	C₂³×C₄
Frattini, cosocle	#494	C₂⁴
socle	C₂	C₂×#13561
prime radical	#6348098	C₂
LCS	G ▷ C₄⋊Q₈ ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄² ◁ #60639 ◁ G
LECS	G ▷ #494 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

2015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=gh,e^2=hij,f^2=j,g^2=i,h^2=j,b^a=befj,c^a=ci,c^b=cij,d^a=dij,d^b=dj,e^b=eg,e^c=ej,e^d=ej,f^a=fgij,f^b=fghj,f^d=fj,f^e=fhij,g^a=gh,g^b=gi,g^f=gi,h^a=hj,h^b=hi,h^f=hj,i^a=ij

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339850, #72339852, #72339864
maximal normal subgroup of	#72339850, #72339852, #72339864

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