#258866

order	512
nilpotency class	3
isoclinism	isoclinic class of #247527
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

740 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)(9,10,15,19)(11,16,12,21)(13,18,14,23)(17,20,24,22)(25,26,29,34)(27,35,37,31)(28,32,30,33)(36,39,38,40);(2,5)(6,7)(9,11,15,12)(10,16,19,21)(13,22,14,20)(17,23,24,18)(25,27)(26,31)(28,38)(29,37)(30,36)(32,39)(33,40)(34,35);(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(26,32)(27,36)(28,30)(33,34)(37,38)(39,40);(1,3,4,8)(2,6,5,7)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)(25,28,29,30)(26,33,34,32)(27,36,37,38)(31,39,35,40)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=ehi,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,d^c=di,f^c=fh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 166062);