#6449802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1596 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)(11,14)(12,13)(15,16)(17,18,21,25)(19,23,29,39)(20,30,32,24)(22,26,34,42)(27,45,47,37)(28,38,35,43)(31,48,46,40)(33,44,36,41);(2,5)(6,8)(17,19,21,29)(18,23,25,39)(20,31,32,46)(22,35,34,28)(24,40,30,48)(26,43,42,38)(27,36,47,33)(37,44,45,41);(10,13)(14,16)(17,20,21,32)(18,24,25,30)(19,27,29,47)(22,36,34,33)(23,37,39,45)(26,44,42,41)(28,46,35,31)(38,48,43,40);(9,11,12,15)(10,14,13,16)(19,28)(22,34)(23,38)(26,42)(27,46)(29,35)(31,47)(33,36)(37,48)(39,43)(40,45)(41,44);(1,3,4,7)(2,6,5,8)(19,29)(20,32)(23,39)(24,30)(28,35)(33,36)(38,43)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6356998);