#350845

order	512
nilpotency class	3
isoclinism	isoclinic class of #346644
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	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

912 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)(11,19)(12,17)(13,18)(14,23)(15,21)(16,20)(22,24)(25,26,28,30)(27,33,34,40)(29,37,36,32)(31,38,39,35);(2,5)(6,8)(9,11)(10,15)(12,22)(13,21)(14,20)(16,17)(18,19)(23,24)(26,30)(27,34)(32,37)(35,38);(1,3,4,7)(2,6,5,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,17)(18,24)(19,23)(29,36)(32,37)(33,40)(35,38);(9,12,13,14)(10,17,18,23)(11,20,21,22)(15,24,19,16)(25,27)(26,31)(28,34)(29,38)(30,39)(32,40)(33,37)(35,36)

Polycyclic presentation:

2^9|a^2=i,c^2=ehi,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,d^b=dhi,f^c=fh,g^c=gi,f^d=fi,g^d=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258041);