#349413

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

797 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=eh,c^2=i,d^2=i,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, 256609);