#7595627

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1542 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,27)(19,26,30,24)(20,32,35,47)(21,37,31,40)(23,38,34,28)(25,41,44,36)(29,42,45,48)(33,39,46,43);(2,5)(6,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,30)(18,24,27,26)(20,33,35,46)(21,34,31,23)(25,42,44,48)(28,37,38,40)(29,36,45,41)(32,43,47,39);(10,14)(11,16)(17,20,34,41,22,35,23,36)(18,25,38,47,27,44,28,32)(19,29,21,33,30,45,31,46)(24,39,37,42,26,43,40,48);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,23,19,22,31,34,30)(18,26,28,37,27,24,38,40)(20,33,36,29,35,46,41,45)(25,43,32,42,44,39,47,48);(2,5)(6,8)(10,14)(13,15)(18,27)(20,35)(24,26)(28,38)(29,45)(33,46)(36,41)(37,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=fgi,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=cgi,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, 7502823);