#6754029

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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

1405 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,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,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6661225);