#6990134

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1347 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=g,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6897330);