Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45105 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,9,11,12)(14,18)(16,19)(20,25)(21,26)(23,28)(24,27);(1,2)(3,4)(5,7)(6,9)(8,11)(10,12)(13,14,19,26,17,21,16,18)(15,20,27,28,22,23,24,25);(6,10)(7,11)(13,15)(14,20)(16,24)(17,22)(18,25)(19,27)(21,23)(26,28);(2,4)(6,10)(9,12)(14,21)(15,22)(18,26)(24,27);(13,14,19,26,17,21,16,18)(15,23,27,25,22,20,24,28)Polycyclic presentation:
2^9|a^2=g,b^2=fi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7434781);