#6820384

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

1320 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)(17,18,22,26)(19,23,29,36)(20,24,32,39)(21,25,33,40)(27,34,42,46)(28,35,31,38)(30,37,43,47)(41,45,44,48);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,14,15,16)(19,27)(20,30)(21,33)(23,34)(24,37)(25,40)(28,31)(29,42)(32,43)(35,38)(36,46)(39,47);(9,12,13,11)(10,14,15,16)(17,19,21,27,22,29,33,42)(18,23,25,34,26,36,40,46)(20,31,30,44,32,28,43,41)(24,38,37,48,39,35,47,45);(1,3,4,7)(2,6,5,8)(20,32)(24,39)(28,31)(30,43)(35,38)(37,47)(41,44)(45,48);(2,5)(6,8)(17,20,33,43,22,32,21,30)(18,24,40,47,26,39,25,37)(19,28,42,44,29,31,27,41)(23,35,46,48,36,38,34,45)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,d^2=f,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^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, 6727580);