#286978

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

573 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)(9,10,13,18,14,19,12,17)(11,15,20,16,22,24,21,23)(25,26,31,37,30,28,29,33)(27,32,38,40,34,36,35,39);(2,5)(6,7)(9,11,12,21,14,22,13,20)(10,15,17,23,19,24,18,16)(25,27,29,35,30,34,31,38)(26,32,33,39,28,36,37,40);(9,11,14,22)(10,16,19,23)(12,20,13,21)(15,17,24,18)(25,28,30,26)(27,32,34,36)(29,33,31,37)(35,40,38,39);(1,3,4,8)(2,6,5,7)(9,12,14,13)(10,17,19,18)(11,20,22,21)(15,16,24,23)(27,34)(32,36)(35,38)(39,40)

Polycyclic presentation:

2^9|a^2=fh,b^2=f,c^2=h,d^2=ehi,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, 194174);