#6825373

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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	#24266	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1525 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)(11,14)(12,13)(15,16)(17,18,22,26)(19,23,28,34)(20,24,32,38)(21,25,31,37)(27,33,39,43)(29,35,41,45)(30,36,42,46)(40,44,47,48);(2,5)(3,7)(19,27)(20,30)(21,31)(23,33)(24,36)(25,37)(28,39)(29,41)(32,42)(34,43)(35,45)(38,46);(10,13)(14,16)(17,19,21,27,22,28,31,39)(18,23,25,33,26,34,37,43)(20,29,30,40,32,41,42,47)(24,35,36,44,38,45,46,48);(2,5)(6,8)(9,11,12,15)(10,14,13,16)(19,28)(23,34)(27,39)(29,41)(33,43)(35,45)(40,47)(44,48);(1,3,4,7)(2,6,5,8)(17,20,31,42,22,32,21,30)(18,24,37,46,26,38,25,36)(19,29,39,47,28,41,27,40)(23,35,43,48,34,45,33,44)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6732569);