#285728

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
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

578 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)(9,10,13,17,15,18,14,11)(12,20,23,16,22,19,21,24)(25,26,30,32,29,33,28,38)(27,34,37,40,36,31,35,39);(2,5)(6,8)(9,11,14,18,15,17,13,10)(12,16,21,20,22,24,23,19)(25,27,30,37,29,36,28,35)(26,31,32,39,33,34,38,40);(1,3,4,7)(2,6,5,8)(9,12,15,22)(10,16,18,24)(11,19,17,20)(13,21,14,23)(25,26)(27,31)(28,32)(29,33)(30,38)(34,36)(35,39)(37,40);(2,5)(6,8)(9,13,15,14)(10,17,18,11)(12,21,22,23)(16,20,24,19)(25,28,29,30)(26,32,33,38)(27,35,36,37)(31,39,34,40)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192924);