#7610073

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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(11,13)(12,14)(15,16)(17,18,23,28)(19,24,31,40)(20,25,32,41)(21,26,35,44)(22,27,37,46)(29,38,36,45)(30,39,34,43)(33,42,47,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,23,31)(18,24,28,40)(20,29,32,36)(21,33,35,47)(22,34,37,30)(25,38,41,45)(26,42,44,48)(27,43,46,39);(1,3,5,4)(2,7,6,8)(10,14)(11,15)(17,20,23,32)(18,25,28,41)(19,29,31,36)(21,34,35,30)(22,33,37,47)(24,38,40,45)(26,43,44,39)(27,42,46,48);(1,4,5,3)(2,8,6,7)(10,14)(13,16)(17,21,22,36,23,35,37,29)(18,26,27,45,28,44,46,38)(19,20,30,33,31,32,34,47)(24,25,39,42,40,41,43,48);(2,6)(7,8)(20,32)(21,35)(25,41)(26,44)(29,36)(33,47)(38,45)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fi,d^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=dg,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, 7517269);