#7615610

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	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1345 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(19,28)(24,29)(30,33)(31,36)(39,42)(40,41)(45,48)(46,47);(2,6)(7,8)(11,16)(13,15)(17,18,22,27)(19,29,28,24)(20,32,35,43)(21,26,23,38)(25,37,44,34)(30,45,33,48)(31,41,36,40)(39,47,42,46);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,19,22,28)(18,24,27,29)(20,33,35,30)(21,36,23,31)(25,42,44,39)(26,41,38,40)(32,45,43,48)(34,46,37,47);(10,14)(13,15)(17,20,21,37,22,35,23,34)(18,25,38,43,27,44,26,32)(19,30,36,46,28,33,31,47)(24,39,40,45,29,42,41,48);(1,4,5,3)(2,7,6,8)(17,21,22,23)(18,26,27,38)(19,31,28,36)(20,34,35,37)(24,40,29,41)(25,43,44,32)(30,46,33,47)(39,48,42,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=fg,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,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, 7522806);