#349314

order	512
nilpotency class	3
isoclinism	isoclinic class of #346898
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	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

943 conjugacy classes of subgroups; only the 164 normal subgroups are shown.

normal subgroups only

This group has 943 conjugacy classes of subgroups — too many to draw. Only its 164 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=ei,b^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,c^b=ch,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, 256510);