#6911422

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1711 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,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, 6818618);