#6804291

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1278 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)(3,7)(4,6)(5,8)(17,18,22,26)(19,27,31,43)(20,24,32,39)(21,35,28,25)(23,36,38,30)(29,42,33,46)(34,48,44,41)(37,47,40,45);(1,3)(2,5)(4,8)(6,7)(9,10,12,13)(11,15,16,14)(20,32)(24,39)(29,33)(34,44)(37,40)(41,48)(42,46)(45,47);(10,13)(14,15)(17,19,28,36,22,31,21,30)(18,23,35,43,26,38,25,27)(20,33,44,45,32,29,34,47)(24,40,48,42,39,37,41,46);(2,6)(5,7)(17,20)(18,24)(19,29)(21,34)(22,32)(23,37)(25,41)(26,39)(27,42)(28,44)(30,45)(31,33)(35,48)(36,47)(38,40)(43,46);(3,8)(5,7)(9,11,12,16)(10,14,13,15)(20,32)(24,39)(29,33)(34,44)(37,40)(41,48)(42,46)(45,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,e^2=h,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=di,e^d=ei

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, 6711487);