#6832503

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1303 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fhi,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, 6739699);