#6808772

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806827
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1632 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,8)(3,9,11,6)(4,12,13,7)(10,14,16,15)(17,18)(19,24)(20,32)(21,35)(22,26)(23,27)(25,38)(28,44)(29,43)(30,33)(31,40)(34,37)(36,48)(39,46)(41,45)(42,47);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,19)(18,24)(20,33)(21,36)(22,29)(23,31)(25,41)(26,43)(27,40)(28,37)(30,32)(34,44)(35,48)(38,45)(39,42)(46,47);(17,20,22,18,23,26,32,27)(19,24,30,33,31,40,29,43)(21,34,37,25,38,42,47,35)(28,39,36,41,46,44,45,48);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15);(3,11)(6,9)(10,16)(14,15)(17,21,32,47,23,38,22,37)(18,25,20,34,27,35,26,42)(19,28,29,45,31,46,30,36)(24,39,43,48,40,44,33,41)

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=f,e^2=hi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,h^b=hi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6715968);