#6615839

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1448 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)(4,8)(5,6)(17,18,23,36,22,26,21,27)(19,28,35,24,31,41,30,39)(20,32,37,40,33,25,34,47)(29,43,48,46,44,42,45,38);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,22,31)(18,24,26,39)(20,29,33,44)(21,35,23,30)(25,38,32,46)(27,41,36,28)(34,48,37,45)(40,43,47,42);(2,6)(7,8)(10,13)(11,15)(17,20)(18,25)(19,29)(21,34)(22,33)(23,37)(24,38)(26,32)(27,40)(28,42)(30,45)(31,44)(35,48)(36,47)(39,46)(41,43);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(20,33)(25,32)(29,44)(34,37)(38,46)(40,47)(42,43)(45,48);(1,4,5,3)(2,8,6,7)(10,13)(11,15)(18,26)(24,39)(25,32)(27,36)(28,41)(38,46)(40,47)(42,43)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6523035);