#6579542

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

Polycyclic presentation:

2^9|a^2=fi,b^2=g,c^2=f,d^2=i,e^2=i,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, 6486738);