#6433152

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432838
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2500 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)(17,18,21,24)(19,27,30,32)(20,26,22,25)(23,31,28,29);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,18)(19,28)(20,25)(21,24)(22,26)(23,30)(27,31)(29,32);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(17,19,21,30)(18,23,24,28)(20,29,22,31)(25,27,26,32);(10,14)(13,16)(18,24)(23,28)(25,26)(27,32);(17,20,21,22)(18,25,24,26)(19,29,30,31)(23,27,28,32)

Polycyclic presentation:

2^9|a^2=i,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6340348);