#6433907

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433057
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2189 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)(19,22,26,24)(20,28,31,27)(25,32,29,30);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,27)(21,22)(23,26)(25,30)(28,31)(29,32);(11,15)(13,16)(17,20,24,30,23,31,22,32)(18,25,26,28,21,29,19,27);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(17,21,23,18)(19,22,26,24)(20,29,31,25)(27,32,28,30);(17,22,23,24)(18,26,21,19)(20,30,31,32)(25,27,29,28)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6341103);