#7618033

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, 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fi,d^2=hi,e^2=g,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, 7525229);