#253218

order	512
nilpotency class	3
isoclinism	isoclinic class of #247367
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

849 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)(9,10,15,11)(12,20,23,24)(13,17,14,19)(16,21,18,22)(27,35)(28,29)(31,39)(33,34)(36,40)(37,38);(1,3,4,8)(2,5,6,7)(9,11,15,10)(12,18,23,16)(13,17,14,19)(20,22,24,21)(25,26)(27,36)(28,34)(29,33)(30,32)(31,38)(35,40)(37,39);(9,12,15,23)(10,16,11,18)(13,22,14,21)(17,24,19,20)(25,27,30,37)(26,31,32,40)(28,38,29,35)(33,36,34,39);(2,6)(5,7)(9,13,15,14)(10,17,11,19)(12,21,23,22)(16,20,18,24)(26,32)(27,37)(33,34)(35,38)

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,c^2=h,d^2=hi,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,d^c=dh,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, 160414);