Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454974 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2487 |
| prime radical | C25 | C2×Q8 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
823 conjugacy classes of subgroups; only the 194 normal subgroups are shown.
This group has 823 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,17,12,18,11,16)(7,19,22,13,24,32,23,33)(8,25,27,29,15,9,28,14)(20,21,26,34,36,30,31,35);(1,2,3,4)(5,7,10,22,12,24,11,23)(6,13,16,19,18,33,17,32)(8,26,27,36,15,31,28,20)(9,21,29,35,25,30,14,34);(2,4)(5,8,11,27,12,15,10,28)(6,14,16,25,18,29,17,9)(7,20,23,26,24,36,22,31)(13,30,19,35,33,21,32,34);(5,9,10,29,12,25,11,14)(6,15,16,28,18,8,17,27)(7,21,22,35,24,30,23,34)(13,31,19,20,33,26,32,36)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=e,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,f^d=fi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 382891);