#6449300

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 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:

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

Polycyclic presentation:

2^9|a^2=f,b^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,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, 6356496);