#323240

order	512
nilpotency class	3
isoclinism	isoclinic class of #323197
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

693 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)(3,9)(4,14)(5,8)(6,11)(7,12)(10,16)(13,15)(17,18,24,27,23,20,22,28)(19,26,21,31,30,32,29,25);(2,8,12,11)(3,13,10,14)(4,15)(9,16)(17,19,24,21,23,30,22,29)(18,25,28,32,20,31,27,26);(1,3)(2,9)(4,11)(5,13)(6,14)(7,10)(8,15)(12,16)(17,20,23,18)(19,25,30,31)(21,32,29,26)(22,28,24,27);(1,4,6,9,7,15,5,16)(2,10,8,13,12,3,11,14)(17,21,22,19,23,29,24,30)(18,26,27,31,20,32,28,25)

Polycyclic presentation:

2^9|a^2=fi,b^2=eg,c^2=i,d^2=ehi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 230436);