#350233

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

922 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)(9,10,14,12)(11,17,18,24)(13,21,23,16)(15,22,19,20)(25,26)(27,35)(28,34)(29,32)(30,33)(31,36)(37,39)(38,40);(2,5)(6,8)(11,18)(15,19)(17,24)(20,22)(25,27)(26,31)(28,36)(29,37)(30,38)(32,40)(33,39)(34,35);(1,3,4,7)(2,6,5,8)(9,11)(10,15)(12,19)(13,22)(14,18)(16,24)(17,21)(20,23)(26,32)(27,36)(29,30)(31,39)(33,34)(35,40);(9,12,14,10)(11,19,18,15)(13,16,23,21)(17,22,24,20)(26,33)(29,30)(31,39)(32,34)(35,40)(37,38)

Polycyclic presentation:

2^9|a^2=hi,c^2=ei,d^2=hi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,c^b=ch,f^c=fh,g^c=gh,f^d=fh,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, 257429);