#286902

order	512
nilpotency class	3
isoclinism	isoclinic class of #282171
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

561 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)(4,5)(6,8)(9,10,13,19,14,17,12,18)(11,15,21,16,22,23,20,24)(25,26,30,34,29,33,28,32)(27,35,38,40,36,31,37,39);(2,5)(6,7)(9,11,12,20,14,22,13,21)(10,15,18,24,17,23,19,16)(25,27,30,38,29,36,28,37)(26,31,34,39,33,35,32,40);(9,11,14,22)(10,16,17,24)(12,21,13,20)(15,18,23,19)(26,32)(27,36)(28,30)(31,39)(33,34)(35,40);(1,3,4,8)(2,6,5,7)(10,17)(15,23)(16,24)(18,19)(25,28,29,30)(26,32,33,34)(27,37,36,38)(31,40,35,39)

Polycyclic presentation:

2^9|a^2=fhi,b^2=f,c^2=hi,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

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, 194098);