#7576096

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1228 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei,h^d=hi

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, 7483292);