#7617146

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

1308 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,15,16,13)(17,18)(19,30)(20,33)(21,38)(22,41)(23,29)(24,28)(25,32)(26,37)(27,40)(31,43)(34,44)(35,45)(36,46)(39,48)(42,47);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19)(18,25)(20,34)(21,22)(23,32)(24,40)(26,44)(27,41)(28,38)(29,30)(31,37)(33,43)(35,42)(36,39)(45,46)(47,48);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20)(18,26)(19,31)(21,39)(22,42)(23,37)(24,36)(25,43)(27,48)(28,47)(29,33)(30,44)(32,34)(35,40)(38,45)(41,46);(1,4,5,3)(2,8,6,7)(10,14)(11,16)(17,21,22,19,23,40,24,32)(18,27,41,25,29,38,28,30)(20,35,42,34,37,39,36,31)(26,45,46,44,33,48,47,43);(2,6)(7,8)(10,14)(13,15)(17,22,23,24)(18,28,29,41)(19,21,32,40)(20,36,37,42)(25,38,30,27)(26,46,33,47)(31,35,34,39)(43,48,44,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=fg,d^2=fh,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, 7524342);