#6883208

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
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	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1393 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)(9,10)(11,15)(12,14)(13,16)(17,18,23,27)(19,24,29,38)(20,32,34,40)(21,26,22,36)(25,35,41,33)(28,42,43,48)(30,39,31,46)(37,45,47,44);(1,3,6,2)(4,8,5,7)(9,11,12,16)(10,13,14,15)(17,19,23,29)(18,24,27,38)(20,28,34,43)(21,30,22,31)(25,37,41,47)(26,39,36,46)(32,42,40,48)(33,44,35,45);(11,16)(13,15)(17,20,21,35,23,34,22,33)(18,25,36,40,27,41,26,32)(19,28,30,45,29,43,31,44)(24,37,46,48,38,47,39,42);(1,4,6,5)(2,7,3,8)(10,14)(13,15)(19,29)(24,38)(28,43)(30,31)(37,47)(39,46)(42,48)(44,45);(1,5,6,4)(2,7,3,8)(17,21,23,22)(18,26,27,36)(19,30,29,31)(20,33,34,35)(24,39,38,46)(25,40,41,32)(28,44,43,45)(37,48,47,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=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, 6790404);