#6875362

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1451 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6782558);