#6449314

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1622 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=g,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,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, 6356510);