#323209

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	#3205	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

673 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)(4,12)(5,8)(6,10)(7,11)(13,15)(14,16)(17,18,22,27,23,20,29,26)(19,28,21,24,30,25,32,31);(2,8,11,10)(3,12,13,16)(4,14)(9,15)(17,19,29,32,23,30,22,21)(18,24,27,25,20,31,26,28);(1,3,7,13)(2,9,11,15)(4,10,14,8)(5,12,6,16)(17,20)(18,23)(19,24)(21,28)(22,27)(25,32)(26,29)(30,31);(1,4,5,9,7,14,6,15)(2,3,10,12,11,13,8,16)(17,21,22,30,23,32,29,19)(18,25,26,24,20,28,27,31)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=hi,d^2=e,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, 230405);