#286868

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

560 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)(9,10,14,16,13,17,12,18)(11,15,19,24,21,22,20,23)(25,26,31,37,29,28,30,33)(27,34,38,39,35,32,36,40);(2,5)(6,7)(9,11,12,20,13,21,14,19)(10,15,18,23,17,22,16,24)(25,27,31,38,29,35,30,36)(26,32,37,40,28,34,33,39);(10,16)(11,19)(12,14)(15,22)(17,18)(20,21)(25,28,29,26)(27,34,35,32)(30,33,31,37)(36,39,38,40);(1,3,4,8)(2,6,5,7)(10,17)(15,22)(16,18)(23,24)

Polycyclic presentation:

2^9|a^2=fh,b^2=fi,c^2=i,d^2=ei,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, 194064);