#351617

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

569 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 569 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,5)(4,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,20)(18,22)(19,21)(23,24)(25,26,30,32)(27,33,36,34)(28,29,37,38)(31,39,40,35);(1,3,4,7)(2,5,6,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,21)(17,23)(20,24)(25,27,29,35,30,36,38,39)(26,31,28,33,32,40,37,34);(2,6)(5,8)(10,15)(11,18)(16,17)(19,24)(26,32)(28,37)(31,40)(33,34);(10,16)(13,20)(14,21)(15,17)(19,24)(22,23)(25,28,29,32,30,37,38,26)(27,34,35,31,36,33,39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=efi,d^2=e,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gi,c^b=chi,d^b=dh,d^c=dh,g^d=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258813);