#6832533

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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

1287 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)(19,24)(20,25)(21,35)(22,27)(23,28)(26,36)(29,37)(30,38)(31,39)(32,40)(33,41)(34,43)(42,46)(44,47)(45,48);(9,11)(10,13)(12,16)(14,15)(17,19,23,30)(18,24,28,38)(20,31,33,46)(21,22,36,32)(25,39,41,42)(26,40,35,27)(29,44,45,34)(37,47,48,43);(2,5)(6,8)(10,14)(11,16)(17,20,32,44,23,33,22,34)(18,25,40,47,28,41,27,43)(19,29,36,46,30,45,21,31)(24,37,26,42,38,48,35,39);(10,14)(13,15)(20,33)(25,41)(29,45)(31,46)(34,44)(37,48)(39,42)(43,47);(1,3,4,7)(2,6,5,8)(17,21,22,30,23,36,32,19)(18,26,27,24,28,35,40,38)(20,31,34,45,33,46,44,29)(25,42,43,37,41,39,47,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6739729);