#253000

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

888 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)(11,17)(12,19)(14,21)(16,23)(18,20)(22,24)(25,26,29,32)(27,30,34,39)(28,36,37,31)(33,40,35,38);(1,3,4,8)(2,5,6,7)(9,10)(11,14)(12,16)(13,15)(17,21)(18,22)(19,23)(20,24)(25,27,29,34)(26,30,32,39)(28,33,37,35)(31,38,36,40);(9,11,13,18)(10,14,15,22)(12,20,19,17)(16,24,23,21)(27,33)(28,37)(30,38)(31,36)(34,35)(39,40);(2,6)(5,7)(10,15)(14,22)(16,23)(21,24)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=h,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, 160196);