#6681528

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1465 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=i,d^2=f,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6588724);