#6820227

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1349 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18)(19,30)(20,26)(21,38)(22,28)(23,29)(24,27)(25,34)(31,47)(32,42)(33,45)(35,48)(36,43)(37,46)(39,41)(40,44);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,25)(20,35)(21,32)(22,37)(23,34)(24,40)(26,28)(27,42)(29,30)(31,36)(33,39)(38,44)(41,43)(45,47)(46,48);(9,12,13,11)(10,14,15,16)(17,20,24,36,23,37,21,33)(18,26,27,43,29,46,38,45)(19,31,32,35,34,39,40,22)(25,41,44,28,30,47,42,48);(2,6)(5,7)(17,21,23,24)(18,27,29,38)(19,32,34,40)(20,36,37,33)(22,39,35,31)(25,42,30,44)(26,45,46,43)(28,47,48,41);(3,8)(5,7)(17,22,24,31,23,35,21,39)(18,28,27,47,29,48,38,41)(19,33,32,20,34,36,40,37)(25,43,44,46,30,45,42,26)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=i,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6727423);