Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499266 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #13341 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
605 conjugacy classes of subgroups; only the 136 normal subgroups are shown.
This group has 605 conjugacy classes of subgroups — too many to draw. Only its 136 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)(7,11)(8,19)(9,14)(10,12)(13,22)(15,31)(16,30)(17,26)(18,25)(20,34)(21,35)(23,27)(24,32)(28,36)(29,33);(7,15)(8,20)(9,23)(11,24)(13,28)(14,27)(16,21)(17,32)(18,33)(19,34)(22,36)(25,35)(26,31)(29,30);(2,4)(5,7)(6,11)(8,21)(9,16)(10,17)(12,26)(13,29)(14,30)(15,20)(18,23)(19,35)(22,33)(24,28)(25,27)(31,34)(32,36);(6,12)(7,16,17,18)(8,22)(11,25,26,30)(14,27)(15,21,32,33)(20,36)(24,35,31,29)Polycyclic presentation:
2^9|a^2=f,d^2=e,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,g^b=gh,h^b=hi,d^c=dh,e^c=ei,g^c=gh,h^c=hi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72250751, #72252775, #72252778, … |
| maximal normal subgroup of | #72250751, #72252775, #72252778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406672);