#6622092

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1419 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)(11,16)(12,15)(13,14)(17,18,23,27)(19,24,29,38)(20,31,32,42)(21,26,22,36)(25,35,41,33)(28,43,46,48)(30,39,34,40)(37,44,47,45);(2,5)(6,8)(10,14)(11,12)(17,19,23,29)(18,24,27,38)(20,28,32,46)(21,34,22,30)(25,37,41,47)(26,40,36,39)(31,43,42,48)(33,45,35,44);(9,11,13,12)(10,15,14,16)(17,20,21,35,23,32,22,33)(18,25,36,42,27,41,26,31)(19,28,34,44,29,46,30,45)(24,37,39,48,38,47,40,43);(1,3,4,7)(2,6,5,8)(11,12)(15,16)(19,29)(20,32)(24,38)(25,41)(30,34)(31,42)(33,35)(39,40);(9,12,13,11)(10,15,14,16)(17,21,23,22)(18,26,27,36)(19,30,29,34)(20,33,32,35)(24,39,38,40)(25,42,41,31)(28,44,46,45)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

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, 6529288);