#286208

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
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

564 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)(9,10,13,11,14,17,15,18)(12,20,24,21,23,19,22,16)(25,26,29,38,31,28,30,33)(27,32,34,40,36,37,35,39);(2,5)(6,8)(9,11,13,17,14,18,15,10)(12,21,24,19,23,16,22,20)(25,27,30,35,31,36,29,34)(26,32,33,39,28,37,38,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,24)(17,21)(18,20)(25,28,31,26)(27,32,36,37)(29,38,30,33)(34,39,35,40);(2,5)(6,8)(9,13,14,15)(10,11,17,18)(12,22,23,24)(16,19,21,20)(25,29,31,30)(26,33,28,38)(27,34,36,35)(32,39,37,40)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=eh,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,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, 193404);