#6843154

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 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)(17,18,23,22)(19,24,29,38)(20,25,31,33)(21,34,37,26)(27,30,39,46)(28,44,41,35)(32,42,36,40)(43,48,45,47);(2,5)(3,7)(17,19,23,29)(18,24,22,38)(20,30,31,46)(21,35,37,44)(25,39,33,27)(26,41,34,28)(32,45,36,43)(40,48,42,47);(10,14)(15,16)(17,20,18,25,23,31,22,33)(19,27,38,46,29,39,24,30)(21,36,26,42,37,32,34,40)(28,45,44,47,41,43,35,48);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(20,31)(25,33)(27,39)(30,46)(32,36)(40,42)(43,45)(47,48);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,16,14,15)(17,21,18,26,23,37,22,34)(19,28,38,44,29,41,24,35)(20,32,25,40,31,36,33,42)(27,43,46,48,39,45,30,47)

Polycyclic presentation:

2^9|a^2=i,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, 6750350);