#7086536

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=h,d^2=f,e^2=gi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6993732);