#265498

order	512
nilpotency class	3
isoclinism	isoclinic class of #248899
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

818 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)(9,10,12,14)(11,17,18,24)(13,21,20,16)(15,22,23,19)(25,26)(27,30)(28,31)(29,32)(33,38)(34,37)(35,39)(36,40);(1,3,5,4)(2,6,8,7)(10,14)(15,23)(16,21)(17,24)(25,27,29,34)(26,30,32,37)(28,33,35,40)(31,38,39,36);(9,11,12,18)(10,15,14,23)(13,22,20,19)(16,24,21,17)(25,28,29,35)(26,31,32,39)(27,33,34,40)(30,36,37,38);(1,4,5,3)(2,7,8,6)(10,14)(11,18)(16,21)(19,22)

Polycyclic presentation:

2^9|a^2=e,b^2=ehi,c^2=hi,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,d^c=dh,f^c=fh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 172694);