#7489762

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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	#42679	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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)(17,18,23,25,21,27,22,28)(19,29,36,38,30,24,32,40)(20,33,37,42,34,26,35,41)(31,39,46,48,44,43,45,47);(2,5)(3,7)(10,14)(11,12)(17,19,21,30)(18,24,27,29)(20,31,34,44)(22,32,23,36)(25,40,28,38)(26,39,33,43)(35,45,37,46)(41,48,42,47);(9,10)(11,16)(12,15)(13,14)(18,25)(19,30)(20,34)(22,23)(24,38)(26,42)(27,28)(29,40)(33,41)(39,48)(43,47)(45,46);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,15,14,16)(17,20)(18,26)(19,31)(21,34)(22,35)(23,37)(24,39)(25,41)(27,33)(28,42)(29,43)(30,44)(32,45)(36,46)(38,47)(40,48);(2,5)(6,8)(9,12,13,11)(10,15,14,16)(18,27)(19,30)(20,34)(25,28)(32,36)(35,37)(39,43)(47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7396958);