#326975

order	512
nilpotency class	3
isoclinism	isoclinic class of #326959
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

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

1273 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)(9,10,12,13)(11,14,16,15)(17,18,22,25)(19,23,21,29)(20,27,28,31)(24,30,32,26);(2,5)(6,8)(9,11)(10,14)(12,16)(13,15)(17,19,22,21)(18,23,25,29)(20,26,28,30)(24,31,32,27);(1,3)(2,6)(4,7)(5,8)(10,13)(11,15)(14,16)(17,20,22,28)(18,24,25,32)(19,26,21,30)(23,31,29,27);(3,7)(6,8)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234171);