#329029

order	512
nilpotency class	3
isoclinism	isoclinic class of #326735
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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 ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

890 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 236225);