#7605165

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1400 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:

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

Polycyclic presentation:

2^9|a^2=fgh,c^2=fhi,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei

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, 7512361);