#6612259

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1660 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)(12,15)(13,16)(17,18,23,27)(19,28,32,45)(20,33,35,25)(21,26,22,38)(24,30,40,31)(29,46,34,44)(36,43,37,42)(39,47,41,48);(9,11)(10,13)(12,16)(14,15)(17,19,22,31,23,32,21,30)(18,24,26,28,27,40,38,45)(20,34,36,48,35,29,37,47)(25,41,43,46,33,39,42,44);(2,5)(6,8)(11,16)(13,14)(17,20)(18,25)(19,29)(21,37)(22,36)(23,35)(24,39)(26,43)(27,33)(28,44)(30,48)(31,47)(32,34)(38,42)(40,41)(45,46);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(20,35)(25,33)(29,34)(36,37)(39,41)(42,43)(44,46)(47,48);(11,16)(13,14)(17,21,23,22)(18,26,27,38)(19,30,32,31)(20,36,35,37)(24,28,40,45)(25,42,33,43)(29,47,34,48)(39,46,41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dh,e^c=ei,e^d=ehi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519455);