Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628892 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
17161 conjugacy classes of subgroups; only the 946 normal subgroups are shown.
This group has 17161 conjugacy classes of subgroups — too many to draw. Only its 946 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 — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,27)(16,22)(17,23)(18,24)(19,25)(20,26)(21,29)(28,35)(30,33)(31,34)(32,36);(5,7)(6,11)(8,15)(9,16)(10,17)(12,21)(13,22)(14,23)(18,28)(19,29)(20,30)(24,32)(25,27)(26,33)(31,36)(34,35);(5,8)(6,12)(7,15)(9,18)(10,19)(11,21)(13,24)(14,25)(16,28)(17,29)(20,31)(22,32)(23,27)(26,34)(30,36)(33,35);(2,4)(5,9,10,20)(6,13,14,26)(7,16,17,30)(8,18,19,31)(11,22,23,33)(12,24,25,34)(15,28,29,36)(21,32,27,35);(6,14)(11,23)(12,25)(13,26)(21,27)(22,33)(24,34)(32,35)Polycyclic presentation:
2^9|d^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^b=gi,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72240597, #72260725, #72265848, … |
| maximal normal subgroup of | #72240597, #72260725, #72265848, … |
| maximal quotient of | #72261560, #72262397 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536089);