#258672

order	512
nilpotency class	3
isoclinism	isoclinic class of #247367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,23,24,22)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35);(2,5)(6,7)(9,11)(10,14)(12,19)(13,20)(15,22)(16,18)(17,23)(21,24)(25,27,30,34)(26,31,29,37)(28,35,36,39)(32,33,38,40);(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(25,28,30,36)(26,32,29,38)(27,33,34,40)(31,39,37,35);(1,3,4,8)(2,6,5,7)(10,16)(11,19)(15,17)(20,24)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,g^b=gi,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, 165868);