#6865526

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 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)(11,15,16,13)(17,18)(19,24)(20,31)(21,26)(22,27)(23,28)(25,29)(30,38)(32,44)(33,37)(34,42)(35,41)(36,40)(39,43)(45,48)(46,47);(1,2)(3,6)(4,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,22)(18,24,28,27)(20,29,33,36)(21,30,35,34)(25,37,40,31)(26,38,41,42)(32,43,46,45)(39,47,48,44);(11,16)(13,15)(17,20,23,33)(18,25,28,40)(19,29,22,36)(21,32,35,46)(24,37,27,31)(26,39,41,48)(30,43,34,45)(38,47,42,44);(1,3,5,4)(2,6,7,8)(10,14)(13,15)(17,21)(18,26)(19,30)(20,32)(22,34)(23,35)(24,38)(25,39)(27,42)(28,41)(29,43)(31,44)(33,46)(36,45)(37,47)(40,48);(1,4,5,3)(2,6,7,8)(17,19,23,22)(18,27,28,24)(20,29,33,36)(21,34,35,30)(25,31,40,37)(26,38,41,42)(32,45,46,43)(39,47,48,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6772722);