#403639

order	512
nilpotency class	3
isoclinism	isoclinic class of #394048
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

1051 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:

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

Polycyclic presentation:

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

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, 310835);