#6820355

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

(2,5)(3,7)(9,10,13,15)(11,14,12,16)(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);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(18,23)(19,28)(20,31)(22,33)(24,37)(25,40)(27,39)(30,43)(32,34)(35,45)(38,41)(42,46);(9,12,13,11)(10,14,15,16)(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);(1,3,4,7)(2,6,5,8)(18,23)(20,32)(24,27)(26,36)(29,44)(31,34)(37,39)(42,46);(2,5)(6,8)(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=g,b^2=gi,c^2=ghi,d^2=fi,e^2=hi,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, 6727551);