#350247

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

844 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)(9,10,15,13)(11,16,14,19)(12,20,23,24)(17,21,22,18)(25,26,29,33)(27,30,35,39)(28,31,36,32)(34,37,40,38);(2,5)(6,8)(9,11,15,14)(10,16,13,19)(12,21,23,18)(17,20,22,24)(25,27,29,35)(26,30,33,39)(28,34,36,40)(31,37,32,38);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,18)(13,22)(14,21)(15,23)(16,24)(19,20);(9,13,15,10)(11,19,14,16)(12,22,23,17)(18,20,21,24)(26,31)(28,36)(30,37)(32,33)(34,40)(38,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=e,d^2=h,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, 257443);