#7063250

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42995	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1515 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=h,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6970446);