#6818250

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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1436 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)(9,10,12,14)(11,15,16,13);(2,5)(3,7)(17,18,22,26)(19,27,28,43)(20,31,35,47)(21,36,33,25)(23,37,38,30)(24,40,42,34)(29,41,32,39)(44,48,46,45);(9,11,12,16)(10,13,14,15)(17,19,21,30,22,28,33,37)(18,23,25,27,26,38,36,43)(20,32,34,48,35,29,40,45)(24,41,31,46,42,39,47,44);(2,5)(6,8)(19,28)(23,38)(27,43)(29,32)(30,37)(39,41)(44,46)(45,48);(1,3,4,7)(2,6,5,8)(17,20,33,40,22,35,21,34)(18,24,36,47,26,42,25,31)(19,29,37,48,28,32,30,45)(23,39,43,46,38,41,27,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,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,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, 6725446);