#349412

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, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

733 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2,4,6)(3,7,8,5)(11,20)(12,22)(13,14)(16,24)(17,18)(21,23)(27,33)(29,37)(31,39)(32,36)(34,40)(35,38);(1,3,4,8)(2,5,6,7)(9,10,15,19)(11,16,21,12)(13,17,14,18)(20,24,23,22)(25,26,30,28)(27,31,35,34)(29,32,37,36)(33,39,38,40);(9,11)(10,16)(12,19)(13,23)(14,20)(15,21)(17,22)(18,24)(25,27,30,35)(26,31,28,34)(29,38,37,33)(32,40,36,39);(9,12,15,16)(10,11,19,21)(13,22,14,24)(17,20,18,23)(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=h,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, 256608);