#6611298

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1434 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)(9,10)(11,14)(12,15)(13,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,30)(29,43)(31,38)(32,40)(33,41)(34,39)(35,42)(37,45)(44,47)(46,48);(9,11)(10,13)(12,16)(14,15)(17,19,21,34,23,31,22,30)(18,24,36,38,27,39,26,28)(20,29,35,44,33,46,32,45)(25,37,40,48,41,47,42,43);(1,3,4,8)(2,5,6,7)(11,16)(13,14)(17,20)(18,25)(19,29)(21,35)(22,32)(23,33)(24,37)(26,42)(27,41)(28,43)(30,45)(31,46)(34,44)(36,40)(38,48)(39,47);(9,11)(10,14)(12,16)(13,15)(17,21,23,22)(18,26,27,36)(19,30,31,34)(20,32,33,35)(24,38,39,28)(25,40,41,42)(29,44,46,45)(37,43,47,48);(11,16)(13,14)(19,31)(20,33)(24,39)(25,41)(28,38)(30,34)(32,35)(40,42)

Polycyclic presentation:

2^9|a^2=g,b^2=fhi,c^2=g,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6518494);