#6830262

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fh,e^2=ghi,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, 6737458);