Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #491097 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#466 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | #19267 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
525 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 525 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.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(10,14)(11,18)(12,19)(15,20)(16,21)(17,24)(22,23);(9,10,15,24,12,17,11,14,13,16,18,23,19,22,20,21);(1,2)(3,6)(4,8)(5,7)(10,16)(14,21)(17,22)(23,24);(1,3,5,4)(2,6,7,8)Polycyclic presentation:
2^9|b^2=egi,d^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,c^b=ci,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72342595, #72342600, #72342606 |
| maximal normal subgroup of | #72342595, #72342600, #72342606 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 398357);