#6794988

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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1758 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,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, 6702184);