#6749218

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1457 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=i,c^2=i,d^2=fi,e^2=f,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, 6656414);