#6455784

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1453 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)(19,27)(20,32)(21,25)(22,26)(23,30)(24,34)(28,38)(29,40)(31,39)(33,41)(35,44)(36,43)(37,42)(45,46)(47,48);(2,5)(6,8)(17,19,21,30)(18,23,25,27)(20,33,34,47)(22,29,36,31)(24,41,32,48)(26,39,43,40)(28,37,46,35)(38,44,45,42);(10,13)(14,16)(17,20,21,34)(18,24,25,32)(19,28,30,46)(22,37,36,35)(23,38,27,45)(26,44,43,42)(29,33,31,47)(39,41,40,48);(1,3,4,7)(2,6,5,8)(19,29)(22,36)(23,39)(26,43)(27,40)(28,33)(30,31)(35,37)(38,41)(42,44)(45,48)(46,47);(9,11,12,15)(10,14,13,16)(18,25)(23,27)(24,32)(26,43)(38,45)(39,40)(41,48)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6362980);