#6883317

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1421 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)(11,15)(12,14)(13,16)(17,18)(19,27)(20,32)(21,23)(22,35)(24,31)(25,33)(26,29)(28,40)(30,44)(34,38)(36,42)(37,48)(39,41)(43,47)(45,46);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(18,23)(19,28)(25,32)(26,35)(30,41)(31,42)(38,47)(45,48);(11,16)(13,15)(17,19,29,42,21,28,22,31)(18,24,35,40,23,36,26,27)(20,30,43,48,33,41,34,45)(25,37,47,44,32,46,38,39);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(17,20)(18,25)(19,30)(21,33)(22,34)(23,32)(24,37)(26,38)(27,39)(28,41)(29,43)(31,45)(35,47)(36,46)(40,44)(42,48);(1,5,6,4)(2,7,3,8)(19,28)(20,33)(24,36)(25,32)(27,40)(31,42)(34,43)(38,47)

Polycyclic presentation:

2^9|b^2=fi,c^2=gi,d^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,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, 6790513);