#6611967

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1719 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,36,22,27,21,26)(19,28,30,24,32,39,31,40)(20,33,37,41,34,25,35,48)(29,43,45,47,44,42,46,38);(9,11,13,12)(10,14,15,16)(17,19,22,32)(18,24,27,40)(20,29,34,44)(21,30,23,31)(25,38,33,47)(26,39,36,28)(35,45,37,46)(41,43,48,42);(2,5)(6,8)(10,15)(11,12)(17,20)(18,25)(19,29)(21,35)(22,34)(23,37)(24,38)(26,41)(27,33)(28,42)(30,45)(31,46)(32,44)(36,48)(39,43)(40,47);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(20,34)(25,33)(29,44)(35,37)(38,47)(41,48)(42,43)(45,46);(11,12)(14,16)(17,21,22,23)(18,26,27,36)(19,30,32,31)(20,35,34,37)(24,39,40,28)(25,41,33,48)(29,45,44,46)(38,43,47,42)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519163);