#266664

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

693 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)(11,12)(13,17)(14,19)(15,20)(16,22)(18,23)(21,24)(25,26)(27,35)(28,34)(29,33)(30,32)(31,37)(36,40)(38,39);(1,3,5,4)(2,6,8,7)(9,11,13,20)(10,15,17,12)(14,21,23,16)(18,24,19,22)(25,27,29,37)(26,31,33,35)(28,36,30,38)(32,40,34,39);(9,12,13,15)(10,16,17,21)(11,18,20,19)(14,24,23,22)(26,32)(28,30)(31,39)(33,34)(35,40)(36,38);(1,4,5,3)(2,7,8,6)(10,17)(12,15)(18,19)(22,24)(25,28,29,30)(26,32,33,34)(27,36,37,38)(31,40,35,39)

Polycyclic presentation:

2^9|a^2=eh,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, 173860);