#6687808

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682648
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#61417	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1647 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=fi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,e^d=ei,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, 6595004);