Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393921 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2305 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 2305 conjugacy classes of subgroups — too many to draw. Only its 282 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(10,15)(12,24)(13,26)(16,31)(18,27)(19,29)(20,36)(21,23)(25,28)(30,33)(32,35)(37,38);(1,2,4,6)(3,7,8,5)(9,10,14,18)(11,19,22,16)(12,17,25,34)(13,27,26,15)(20,35,38,32)(21,31,23,29)(24,30,28,33)(36,40,37,39);(1,3,4,8)(2,5,6,7)(9,11,14,22)(10,16,18,19)(12,20,25,38)(13,23,26,21)(15,29,27,31)(17,32,34,35)(24,36,28,37)(30,39,33,40);(9,12,14,25)(10,17,18,34)(11,20,22,38)(13,28,26,24)(15,30,27,33)(16,32,19,35)(21,36,23,37)(29,39,31,40)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,e^c=ei,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310545);