#7586686

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

1224 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=gi,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,d^c=di,e^c=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, 7493882);