#286728

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	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

572 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)(9,10,14,16,12,17,13,18)(11,19,20,23,21,15,22,24)(25,26,31,37,30,28,29,33)(27,32,38,40,35,36,34,39);(2,5)(6,7)(9,11,13,22,12,21,14,20)(10,15,18,23,17,19,16,24)(25,27,31,38,30,35,29,34)(26,32,37,40,28,36,33,39);(10,16)(11,20)(13,14)(15,19)(17,18)(21,22)(25,28,30,26)(27,32,35,36)(29,33,31,37)(34,40,38,39);(1,3,4,8)(2,6,5,7)(10,17)(15,19)(16,18)(23,24)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=i,d^2=ehi,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, 193924);