#312114

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=hi,d^2=ef,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gi,e^d=eh,f^d=fh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 219310);