#265709

order	512
nilpotency class	3
isoclinism	isoclinic class of #245999
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	C₂³×C₄	C₂⁴
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

874 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=i,d^2=ei,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,f^c=fh,g^c=gh

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, 172905);