#6680395

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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

1441 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)(11,13)(12,14)(15,16)(17,18)(19,28)(20,23)(21,33)(22,34)(24,27)(25,31)(26,36)(29,38)(30,39)(32,41)(35,40)(37,47)(42,45)(43,46)(44,48);(2,5)(3,7)(9,11,12,15)(10,13,14,16)(17,19,23,31)(18,25,20,28)(21,30,36,45)(22,32,24,38)(26,39,33,42)(27,41,34,29)(35,46,37,48)(40,44,47,43);(10,14)(13,16)(17,20,23,18)(19,29,31,41)(21,26,36,33)(22,34,24,27)(25,38,28,32)(30,44,45,43)(35,40,37,47)(39,46,42,48);(2,5)(6,8)(17,21,23,36)(18,26,20,33)(19,30,31,45)(22,35,24,37)(25,39,28,42)(27,40,34,47)(29,43,41,44)(32,46,38,48);(1,3,4,7)(2,6,5,8)(17,22)(18,27)(19,31)(20,34)(21,35)(23,24)(26,40)(29,41)(30,45)(33,47)(36,37)(43,44)

Polycyclic presentation:

2^9|b^2=g,c^2=i,d^2=i,e^2=fi,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,h^c=hi,h^e=hi

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, 6587591);