#350235

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

877 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)(9,10,14,11)(12,19,21,24)(13,18,23,16)(15,22,17,20)(25,26,29,32)(27,33,35,40)(28,36,38,31)(30,37,39,34);(2,5)(6,8)(9,11,14,10)(12,15,21,17)(13,18,23,16)(19,22,24,20);(1,3,4,7)(2,6,5,8)(9,12)(10,15)(11,17)(13,22)(14,21)(16,24)(18,19)(20,23);(13,23)(16,18)(19,24)(20,22)(25,27,29,35)(26,30,32,39)(28,37,38,34)(31,40,36,33)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=e,d^2=i,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, 257431);