#464662

order	512
nilpotency class	3
isoclinism	isoclinic class of #464662
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16508
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 371858);