#350040

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

930 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)(3,6)(4,5)(7,8)(9,10,15,11)(12,20,22,24)(13,18,14,17)(16,23,19,21)(25,26,29,32)(27,33,35,40)(28,36,38,31)(30,37,39,34);(2,5)(6,8)(9,11,15,10)(12,19,22,16)(13,18,14,17)(20,21,24,23);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,23)(14,21)(15,22)(17,24)(18,20)(25,27)(26,30)(28,37)(29,35)(31,40)(32,39)(33,36)(34,38);(9,12)(10,16)(11,19)(13,21)(14,23)(15,22)(17,20)(18,24)(28,38)(31,36)(33,40)(34,37)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257236);