Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476144 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2675 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1297 conjugacy classes of subgroups; only the 178 normal subgroups are shown.
This group has 1297 conjugacy classes of subgroups — too many to draw. Only its 178 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:
(1,2)(3,4)(5,6,10,14,11,15,24,34)(7,16,19,29,20,32,26,12)(8,13,22,31,9,23,25,33)(17,28,35,30,21,36,18,27);(2,3)(5,7)(6,12)(8,21)(9,17)(10,26)(11,20)(13,30)(14,32)(15,29)(16,34)(18,25)(19,24)(22,35)(23,27)(28,31)(33,36);(1,3,4,2)(5,8,10,22,11,9,24,25)(6,13,14,31,15,23,34,33)(7,17,19,35,20,21,26,18)(12,27,16,28,29,30,32,36);(2,3)(5,9)(6,13)(7,18)(8,11)(10,22)(12,28)(14,33)(15,23)(16,27)(17,26)(19,21)(20,35)(24,25)(29,36)(30,32)(31,34)Polycyclic presentation:
2^9|a^2=ef,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,e^b=ei,h^b=hi,d^c=de,g^c=gi,e^d=ei,g^d=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383685);