#6912761

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907977
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×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1624 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=g,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh,e^c=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6819957);