#243134

order	512
nilpotency class	3
isoclinism	isoclinic class of #243130
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

598 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:

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

Polycyclic presentation:

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

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, 150330);