#6842225

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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1301 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)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,18,21,23)(19,27,30,24)(20,25,32,35)(22,26,33,36)(28,37,43,39)(29,41,44,38)(31,45,34,40)(42,47,46,48);(9,11,12,16)(10,13,14,15)(18,23)(19,28)(20,31)(22,33)(24,37)(25,40)(27,39)(30,43)(32,34)(35,45)(38,41)(42,46);(2,6)(7,8)(10,14)(11,16)(17,19,22,28,21,30,33,43)(18,24,26,39,23,27,36,37)(20,29,34,42,32,44,31,46)(25,38,40,48,35,41,45,47);(1,3,5,4)(2,7,6,8)(10,14)(13,15)(18,23)(20,32)(24,27)(26,36)(29,44)(31,34)(37,39)(42,46);(1,4,5,3)(2,8,6,7)(17,20,33,31,21,32,22,34)(18,25,36,45,23,35,26,40)(19,29,43,46,30,44,28,42)(24,38,37,47,27,41,39,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fhi,d^2=fg,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,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, 6749421);