#6820354

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×D₈	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1318 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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, 6727550);