#317860

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

576 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:

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

Polycyclic presentation:

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

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, 225056);