#6452716

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

1647 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)(11,16)(12,14)(13,15)(17,18,21,25)(19,27,30,23)(20,32,34,24)(22,26,36,43)(28,38,47,45)(29,40,31,39)(33,41,46,48)(35,44,37,42);(2,5)(6,8)(17,19,21,30)(18,23,25,27)(20,33,34,46)(22,29,36,31)(24,41,32,48)(26,39,43,40)(28,37,47,35)(38,44,45,42);(9,11,13,12)(10,14,15,16)(17,20,21,34)(18,24,25,32)(19,28,30,47)(22,37,36,35)(23,38,27,45)(26,44,43,42)(29,33,31,46)(39,41,40,48);(9,12,13,11)(10,14,15,16)(19,29)(20,34)(22,36)(23,39)(24,32)(26,43)(27,40)(28,46)(30,31)(33,47)(38,48)(41,45);(1,3,4,7)(2,6,5,8)(20,34)(24,32)(28,47)(33,46)(35,37)(38,45)(41,48)(42,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,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, 6359912);