#6909331

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1483 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)(3,7,8,5)(17,18,21,23)(19,27,29,42)(20,31,32,25)(22,36,35,26)(24,38,37,30)(28,44,33,43)(34,41,45,47)(39,46,40,48);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(18,23)(20,32)(24,37)(26,36)(27,42)(28,33)(34,45)(46,48);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,22,30,21,29,35,38)(18,24,26,27,23,37,36,42)(20,33,34,48,32,28,45,46)(25,40,41,44,31,39,47,43);(2,6)(5,7)(11,16)(13,15)(17,20,21,32)(18,25,23,31)(19,28,29,33)(22,34,35,45)(24,39,37,40)(26,41,36,47)(27,43,42,44)(30,46,38,48);(10,14)(13,15)(20,32)(25,31)(28,33)(34,45)(39,40)(41,47)(43,44)(46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6816527);