#7617004

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1302 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)(9,10,12,14)(11,15,16,13)(17,18,23,28)(19,24,31,39)(20,25,34,42)(21,26,22,27)(29,37,32,40)(30,38,36,44)(33,41,35,43)(45,47,46,48);(2,5)(6,8)(9,11,12,16)(10,13,14,15)(17,19,23,31)(18,24,28,39)(20,29,34,32)(21,30,22,36)(25,37,42,40)(26,38,27,44)(33,45,35,46)(41,47,43,48);(1,3,4,7)(2,6,5,8)(10,14)(11,16)(17,20,23,34)(18,25,28,42)(19,29,31,32)(21,35,22,33)(24,37,39,40)(26,43,27,41)(30,46,36,45)(38,48,44,47);(10,14)(13,15)(17,19,22,30,23,31,21,36)(18,24,27,38,28,39,26,44)(20,32,33,45,34,29,35,46)(25,40,41,47,42,37,43,48);(2,5)(6,8)(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,30,31,36)(20,33,34,35)(24,38,39,44)(25,41,42,43)(29,45,32,46)(37,47,40,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fgi,d^2=h,e^2=i,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, 7524200);