#6827217

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806839
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1406 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)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,24)(20,30)(21,23)(22,26)(25,33)(27,40)(28,42)(29,36)(31,41)(32,39)(34,37)(35,45)(38,44)(43,47)(46,48);(9,11,14,10)(12,16,13,15)(18,23)(19,27)(20,31)(22,34)(24,38)(25,41)(29,44)(30,45)(33,35)(36,40)(39,42)(43,46);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(17,19,22,27,21,29,34,44)(18,24,26,40,23,36,37,38)(20,32,35,46,33,28,31,43)(25,42,41,47,30,39,45,48);(1,3,4,7)(2,6,5,8)(9,12,14,13)(10,15,11,16)(18,23)(20,33)(24,36)(26,37)(28,32)(31,35)(38,40)(43,46);(9,13,14,12)(10,15,11,16)(17,20,34,31,21,33,22,35)(18,25,37,45,23,30,26,41)(19,28,44,46,29,32,27,43)(24,39,38,47,36,42,40,48)

Polycyclic presentation:

2^9|b^2=fi,c^2=fhi,d^2=g,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=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, 6734413);