#7597742

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fh,d^2=g,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^c=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, 7504938);