#7594591

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1372 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=h,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,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, 7501787);