#7615222

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	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1288 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)(4,5)(6,8)(9,10,12,14)(11,15,16,13)(17,18)(19,25)(20,34)(21,39)(22,28)(23,29)(24,30)(26,38)(27,32)(31,44)(33,43)(35,42)(36,45)(37,48)(40,47)(41,46);(2,5)(6,7)(11,16)(13,15)(17,19,23,33)(18,25,29,43)(20,35,38,31)(21,22,32,24)(26,44,34,42)(27,30,39,28)(36,41,40,37)(45,46,47,48);(9,11,12,16)(10,13,14,15)(17,20,23,38)(18,26,29,34)(19,31,33,35)(21,40,32,36)(22,41,24,37)(25,42,43,44)(27,47,39,45)(28,48,30,46);(10,14)(13,15)(17,21,24,19,23,32,22,33)(18,27,30,43,29,39,28,25)(20,36,37,35,38,40,41,31)(26,45,46,42,34,47,48,44);(1,3,4,8)(2,6,5,7)(17,22,23,24)(18,28,29,30)(19,32,33,21)(20,37,38,41)(25,27,43,39)(26,46,34,48)(31,40,35,36)(42,45,44,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=g,d^2=h,e^2=fi,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, 7522418);