#6622255

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574270
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

1388 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:

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

Polycyclic presentation:

2^9|a^2=f,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,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6529451);