Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13694 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D16 |
| commutator, Frattini, cosocle | C2×C16 | C24 |
| socle | C22 | C22×D16 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
450 conjugacy classes of subgroups; only the 142 normal subgroups are shown.
This group has 450 conjugacy classes of subgroups — too many to draw. Only its 142 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6)(7,18)(8,14)(9,27)(10,31)(11,33)(12,17)(13,21)(15,19)(16,28)(20,40)(22,53)(23,55)(24,43)(25,56)(26,39)(29,59)(30,58)(32,60)(34,48)(35,62)(36,52)(37,45)(38,47)(41,44)(42,50)(46,57)(49,65)(51,61)(54,68)(63,66)(64,67);(5,7,19,44,10,22,25,48,8,20,45,66,26,49,30,52,12,24,47,67,32,54,29,51,11,23,46,62,28,50,9,21)(6,13,27,42,16,35,57,55,33,61,59,68,60,64,38,43,17,36,58,65,39,63,37,40,14,34,56,53,31,41,15,18);(2,4);(5,8,12,11)(6,14,17,33)(7,20,24,23)(9,25,30,29)(10,26,32,28)(13,34,36,61)(15,37,38,57)(16,31,39,60)(18,40,43,55)(19,45,47,46)(21,48,52,51)(22,49,54,50)(27,56,58,59)(35,41,63,64)(42,53,65,68)(44,66,67,62)Polycyclic presentation:
2^9|b^2=eg,d^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,g^a=gh,h^a=hi140 conjugacy classes — view the full character table.
| maximal subgroup of | #72234362, #72286857, #72286862, … |
| maximal normal subgroup of | #72234362, #72286857, #72286862, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 420130);