#6612105

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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1671 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)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,28,23,26,21,27)(19,29,31,38,32,42,36,24)(20,25,34,39,33,40,35,41)(30,43,46,48,44,47,45,37);(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,30)(21,31)(22,36)(23,32)(25,37)(26,38)(27,29)(28,42)(33,44)(34,45)(35,46)(39,47)(40,48)(41,43);(2,5)(6,8)(10,15)(11,12)(17,20,23,33)(18,25,26,40)(19,30,32,44)(21,35,22,34)(24,37,38,48)(27,41,28,39)(29,43,42,47)(31,46,36,45);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,15,14)(18,26)(20,33)(24,38)(27,28)(29,42)(30,44)(34,35)(45,46);(11,12)(14,16)(17,21,23,22)(18,27,26,28)(19,31,32,36)(20,34,33,35)(24,29,38,42)(25,39,40,41)(30,45,44,46)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,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, 6519301);