#252972

order	512
nilpotency class	3
isoclinism	isoclinic class of #247527
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

903 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)(4,6)(5,8)(9,10,12,16)(11,18,19,14)(13,15,21,17)(20,23,24,22)(28,36)(30,39)(32,37)(34,38);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,19)(13,20)(15,22)(16,18)(17,23)(21,24)(25,26,29,31)(27,33,35,40)(28,37,36,32)(30,38,39,34);(10,15)(13,21)(14,22)(16,17)(18,23)(20,24)(25,27,29,35)(26,30,31,39)(28,38,36,34)(32,40,37,33);(2,6)(5,7)(10,16)(11,19)(15,17)(20,24)

Polycyclic presentation:

2^9|a^2=h,b^2=ehi,c^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=gi,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, 160168);