#266658

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

718 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)(9,10,15,11)(12,20,22,21)(13,19,14,17)(16,24,18,23)(25,26,30,34)(27,31,36,40)(28,33,29,32)(35,38,37,39);(1,3,5,4)(2,6,8,7)(9,11,15,10)(12,21,22,20)(13,19,14,17)(16,24,18,23)(25,27,30,36)(26,31,34,40)(28,35,29,37)(32,39,33,38);(9,12,15,22)(10,16,11,18)(13,24,14,23)(17,21,19,20)(26,32)(28,29)(31,38)(33,34)(35,37)(39,40);(1,4,5,3)(2,7,8,6)(12,22)(16,18)(20,21)(23,24)(25,28,30,29)(26,33,34,32)(27,35,36,37)(31,38,40,39)

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=hi,d^2=ei,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, 173854);