#6914622

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1467 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)(3,6)(4,8)(5,7)(17,18,23,27)(19,28,29,44)(20,25,35,43)(21,26,22,37)(24,31,38,32)(30,45,33,48)(34,41,36,42)(39,47,40,46);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(19,29)(24,38)(28,44)(30,33)(31,32)(39,40)(45,48)(46,47);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,31,23,29,22,32)(18,24,37,44,27,38,26,28)(20,33,36,46,35,30,34,47)(25,40,41,45,43,39,42,48);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,39)(26,42)(27,43)(28,45)(29,33)(31,47)(32,46)(37,41)(38,40)(44,48);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,31,29,32)(20,34,35,36)(24,28,38,44)(25,41,43,42)(30,46,33,47)(39,48,40,45)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,d^2=f,e^2=fi,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, 6821818);