#7594694

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=h,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,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, 7501890);