Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #36652 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2098 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,21,18,22,29,36)(15,19,25,32,24,31,39,43)(16,20,28,35,27,34,37,41)(23,30,38,42,26,33,40,44);(5,7)(6,9)(8,12)(10,11)(13,15,17,25,18,24,29,39)(14,19,21,32,22,31,36,43)(16,26,28,40,27,23,37,38)(20,33,35,44,34,30,41,42);(2,4)(13,16,17,28,18,27,29,37)(14,20,21,35,22,34,36,41)(15,23,25,38,24,26,39,40)(19,30,32,42,31,33,43,44);(7,12)(9,11)(15,24)(16,27)(19,31)(20,34)(25,39)(28,37)(32,43)(35,41);(6,10)(9,11)(16,27)(20,34)(23,26)(28,37)(30,33)(35,41)(38,40)(42,44)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,d^c=di,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6301895);