#266663

order	512
nilpotency class	3
isoclinism	isoclinic class of #246847
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

697 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=hi,d^2=ehi,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dh,f^c=fh,g^c=ghi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 173859);