Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #97058 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1021 conjugacy classes of subgroups; only the 146 normal subgroups are shown.
This group has 1021 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 52 points:
(5,6)(7,13,15,10)(8,11)(9,12)(14,19,20,17)(16,18)(21,22,25,29,26,30,36,42)(23,27,33,39,32,38,46,50)(24,28,35,41,34,40,47,51)(31,43,45,52,44,37,48,49);(1,2)(3,4)(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,27)(24,31)(25,33)(26,32)(28,37)(29,39)(30,38)(34,44)(35,45)(36,46)(40,43)(41,49)(42,50)(47,48)(51,52);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24)(22,28)(23,31)(25,35)(26,34)(27,37)(29,41)(30,40)(32,44)(33,45)(36,47)(38,43)(39,49)(42,51)(46,48)(50,52)Polycyclic presentation:
2^9|a^2=g,c^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=ei,g^b=gh,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22549);