#349933

order	512
nilpotency class	3
isoclinism	isoclinic class of #346918
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

781 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 781 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=e,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,d^b=dh,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, 257129);