Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2397 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
978 conjugacy classes of subgroups; only the 370 normal subgroups are shown.
This group has 978 conjugacy classes of subgroups — too many to draw. Only its 370 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) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,16)(8,22)(9,26)(10,14)(11,15)(12,19)(13,25)(17,20)(18,31)(21,28)(23,29)(24,36)(27,34)(30,33)(32,35);(5,7,9,19,11,20,10,21)(6,12,14,16,15,28,26,17)(8,18,23,32,25,33,24,34)(13,27,29,30,22,35,36,31);(5,6)(7,17,20,16)(8,13)(9,26)(10,14)(11,15)(12,19,28,21)(18,31,33,30)(22,25)(23,36)(24,29)(27,32,35,34);(5,8,10,24,11,25,9,23)(6,13,14,29,15,22,26,36)(7,18,21,34,20,33,19,32)(12,27,16,30,28,35,17,31)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,d^2=g,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,c^b=cg,f^b=fi,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266183);