#6822147

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1298 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=gi,e^2=h,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, 6729343);