#6612191

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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

1651 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)(19,28,31,40)(20,32,34,25)(21,26,22,38)(24,36,41,30)(29,46,33,45)(35,44,37,43)(39,48,42,47);(9,11)(10,13)(12,16)(14,15)(17,19,21,36,23,31,22,30)(18,24,38,40,27,41,26,28)(20,33,37,48,34,29,35,47)(25,42,43,45,32,39,44,46);(2,5)(6,8)(11,16)(13,14)(17,20)(18,25)(19,29)(21,37)(22,35)(23,34)(24,39)(26,44)(27,32)(28,45)(30,48)(31,33)(36,47)(38,43)(40,46)(41,42);(1,3,4,7)(2,6,5,8)(9,11)(10,14)(12,16)(13,15)(20,34)(25,32)(29,33)(35,37)(39,42)(43,44)(45,46)(47,48);(11,16)(13,14)(17,21,23,22)(18,26,27,38)(19,30,31,36)(20,35,34,37)(24,40,41,28)(25,43,32,44)(29,47,33,48)(39,45,42,46)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6519387);