Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #354 |
| commutator | C42 | C4×C8 |
| Frattini, cosocle | C2×#423 | C22 |
| socle | C22 | #1319 |
| prime radical | C24 | #100 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C23×C4 ◁ C2×#423 ◁ G | |
| LECS | G ▷ C2×#423 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
292 conjugacy classes of subgroups; only the 66 normal subgroups are shown.
This group has 292 conjugacy classes of subgroups — too many to draw. Only its 66 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 — 2 generators on 36 points:
(5,6,8,13,9,14,23,31,10,15,21,32,24,12,22,29)(7,16,18,33,19,27,20,36,17,28,34,30,35,11,25,26);(1,2,3,4)(5,7)(6,11)(8,20,21,25)(9,19)(10,17)(12,28)(13,30,32,33)(14,16)(15,27)(18,23,34,22)(24,35)(26,29,36,31)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=hi,d^2=g,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fh,h^a=hi,c^b=ci,d^b=dfh,d^c=dhi,e^d=ei,f^d=fi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1308);