#6449424

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,30)(21,24)(22,25)(26,27)(28,31)(29,32);(2,5)(6,8)(17,19,22,29)(18,23,25,32)(20,28,21,26)(24,27,30,31);(1,3,4,7)(2,6,5,8)(10,13)(14,16)(19,26)(20,21)(23,27)(24,30)(28,29)(31,32);(9,11,12,15)(10,14,13,16)(17,18)(19,27)(20,24)(21,30)(22,25)(23,26)(28,32)(29,31);(17,20,22,21)(18,24,25,30)(19,26,29,28)(23,31,32,27)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,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, 6356620);