#6843155

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1366 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)(17,18,22,23)(19,24,29,38)(20,30,32,25)(21,34,37,26)(27,43,39,31)(28,45,41,35)(33,40,36,42)(44,47,46,48);(2,5)(3,7)(17,19,22,29)(18,24,23,38)(20,31,32,43)(21,35,37,45)(25,39,30,27)(26,41,34,28)(33,46,36,44)(40,48,42,47);(10,14)(15,16)(17,20,18,25,22,32,23,30)(19,27,38,43,29,39,24,31)(21,36,26,42,37,33,34,40)(28,46,45,47,41,44,35,48);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(20,32)(25,30)(27,39)(31,43)(33,36)(40,42)(44,46)(47,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,18,26,22,37,23,34)(19,28,38,45,29,41,24,35)(20,33,25,40,32,36,30,42)(27,44,43,48,39,46,31,47)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=gi,e^2=fgh,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, 6750351);