Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417903 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2684 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#127 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1280 conjugacy classes of subgroups; only the 186 normal subgroups are shown.
This group has 1280 conjugacy classes of subgroups — too many to draw. Only its 186 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,8,14,10,11,9,13)(7,15,16,19,17,12,18,20);(6,11)(7,16)(8,9)(12,19)(15,20)(17,18);(5,7,8,16,10,17,9,18)(6,12,13,19,11,15,14,20);(2,4)(6,13,11,14)(12,19,15,20)Polycyclic presentation:
2^9|a^2=g,c^2=eg,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72279625, #72338616, #72338620, … |
| maximal normal subgroup of | #72279625, #72338616, #72338620, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 351810);