#446493

order	512
nilpotency class	3
isoclinism	isoclinic class of #417575
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 353689);