#7598335

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

1415 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)(17,18,24,29)(19,25,32,42)(20,33,36,45)(21,39,31,41)(22,28,23,27)(26,38,44,37)(30,43,46,48)(34,40,35,47);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,24,32)(18,25,29,42)(20,34,36,35)(21,23,31,22)(26,43,44,48)(27,41,28,39)(30,38,46,37)(33,40,45,47);(10,13)(14,15)(17,20,23,38,24,36,22,37)(18,26,28,45,29,44,27,33)(19,30,21,35,32,46,31,34)(25,40,41,48,42,47,39,43);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,21,23,32,24,31,22,19)(18,25,28,41,29,42,27,39)(20,35,38,46,36,34,37,30)(26,40,45,48,44,47,33,43);(1,4,5,3)(2,7,6,8)(20,36)(26,44)(30,46)(33,45)(34,35)(37,38)(40,47)(43,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=h,d^2=fgi,e^2=f,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, 7505531);