#285917

order	512
nilpotency class	3
isoclinism	isoclinic class of #281787
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	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

554 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)(3,6)(4,5)(7,8)(9,10,14,16,13,18,12,17)(11,15,19,24,20,22,21,23)(25,26,29,37,30,28,31,33)(27,34,38,40,36,32,35,39);(2,5)(6,8)(9,11,12,21,13,20,14,19)(10,15,17,23,18,22,16,24)(25,27,31,35,30,36,29,38)(26,32,33,40,28,34,37,39);(1,3,4,7)(2,6,5,8)(10,16)(11,19)(12,14)(15,22)(17,18)(20,21)(25,28,30,26)(27,34,36,32)(29,37,31,33)(35,40,38,39);(2,5)(6,8)(11,20)(15,22)(19,21)(23,24)(25,29,30,31)(26,33,28,37)(27,35,36,38)(32,39,34,40)

Polycyclic presentation:

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

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, 193113);