#6820143

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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1362 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18)(19,24)(20,25)(21,26)(22,38)(23,28)(27,34)(29,39)(30,40)(31,45)(32,42)(33,43)(35,44)(36,41)(37,46)(47,48);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,23,33)(18,24,28,43)(20,34,37,22)(21,30,35,32)(25,27,46,38)(26,40,44,42)(29,31,48,36)(39,45,47,41);(9,12,13,11)(10,14,15,16)(17,20,35,31,23,37,21,36)(18,25,44,45,28,46,26,41)(19,29,30,22,33,48,32,34)(24,39,40,38,43,47,42,27);(2,6)(5,7)(17,21,23,35)(18,26,28,44)(19,30,33,32)(20,31,37,36)(22,29,34,48)(24,40,43,42)(25,45,46,41)(27,47,38,39);(3,8)(5,7)(17,22,21,29,23,34,35,48)(18,27,26,47,28,38,44,39)(19,31,32,20,33,36,30,37)(24,41,42,46,43,45,40,25)

Polycyclic presentation:

2^9|a^2=g,b^2=gi,c^2=ghi,d^2=i,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, 6727339);