#6441165

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
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	#23393	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2030 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6348361);