#6843138

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1382 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)(11,15)(12,16)(13,14)(20,32)(24,40)(30,34)(33,46)(35,41)(39,42)(44,47)(45,48);(2,5)(3,7)(17,18,22,26)(19,27,29,43)(20,33,32,46)(21,36,28,25)(23,37,38,31)(24,41,40,35)(30,42,34,39)(44,48,47,45);(10,14)(15,16)(17,19,28,37,22,29,21,31)(18,23,36,43,26,38,25,27)(20,34,41,45,32,30,35,48)(24,42,46,44,40,39,33,47);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(19,29)(23,38)(27,43)(30,34)(31,37)(39,42)(44,47)(45,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,20,28,41,22,32,21,35)(18,24,36,46,26,40,25,33)(19,30,37,48,29,34,31,45)(23,39,43,47,38,42,27,44)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=g,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=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, 6750334);