#6580140

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(11,13)(12,14)(15,16)(17,18)(19,25)(20,24)(21,26)(22,27)(23,30)(28,44)(29,40)(31,39)(32,48)(33,41)(34,42)(35,43)(36,45)(37,46)(38,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,21,30)(18,23,26,25)(20,32,34,33)(22,29,36,31)(24,41,42,48)(27,39,45,40)(28,37,47,35)(38,46,44,43);(10,14)(13,16)(17,20,21,34)(18,24,26,42)(19,28,30,47)(22,37,36,35)(23,38,25,44)(27,46,45,43)(29,32,31,33)(39,41,40,48);(1,3,4,7)(2,6,5,8)(17,19,21,30)(18,25,26,23)(20,33,34,32)(22,29,36,31)(24,41,42,48)(27,40,45,39)(28,35,47,37)(38,46,44,43);(19,29)(22,36)(23,39)(25,40)(27,45)(28,32)(30,31)(33,47)(35,37)(38,41)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=fi,b^2=gi,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6487336);