#7616278

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 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)(11,15)(12,14)(13,16)(17,18)(19,30)(20,34)(21,26)(22,29)(23,28)(24,33)(25,38)(27,41)(31,42)(32,43)(35,44)(36,47)(37,46)(39,48)(40,45);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19,23,33)(18,24,28,30)(20,35,38,31)(21,22,32,41)(25,44,34,42)(26,27,43,29)(36,39,40,37)(45,48,47,46);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20,23,38)(18,25,28,34)(19,31,33,35)(21,40,32,36)(22,39,41,37)(24,42,30,44)(26,47,43,45)(27,48,29,46);(1,4,5,3)(2,8,6,7)(10,14)(11,16)(17,21,22,33,23,32,41,19)(18,26,29,24,28,43,27,30)(20,36,39,31,38,40,37,35)(25,45,46,44,34,47,48,42);(2,6)(7,8)(10,14)(13,15)(17,22,23,41)(18,27,28,29)(19,32,33,21)(20,37,38,39)(24,43,30,26)(25,46,34,48)(31,40,35,36)(42,47,44,45)

Polycyclic presentation:

2^9|b^2=g,c^2=fgi,d^2=fhi,e^2=i,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, 7523474);