#6830254

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	#42883	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1541 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)(12,14)(13,16)(17,18,22,26)(19,29,31,24)(20,32,34,25)(21,27,37,39)(23,28,36,43)(30,38,46,44)(33,40,48,41)(35,47,45,42);(9,11)(10,13)(12,16)(14,15)(17,19,22,31)(18,24,26,29)(20,33,34,48)(21,23,37,36)(25,40,32,41)(27,43,39,28)(30,45,46,35)(38,47,44,42);(2,5)(6,8)(10,14)(11,16)(17,20,23,35,22,34,36,45)(18,25,28,42,26,32,43,47)(19,30,21,33,31,46,37,48)(24,38,39,40,29,44,27,41);(10,14)(13,15)(18,26)(20,34)(24,29)(27,39)(28,43)(30,46)(33,48)(35,45);(1,3,4,7)(2,6,5,8)(17,21,36,19,22,37,23,31)(18,27,43,29,26,39,28,24)(20,33,45,30,34,48,35,46)(25,41,47,44,32,40,42,38)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fhi,e^2=gh,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, 6737450);