Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | #45387 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2107 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14,18,22)(15,23,27,40)(16,28,31,43)(17,32,25,21)(19,33,35,24)(20,36,38,29)(26,37,30,34)(39,44,42,41);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(15,24)(16,29)(17,25)(19,23)(20,28)(21,32)(26,30)(27,33)(31,36)(34,37)(35,40)(38,43);(6,9)(10,11)(13,15,25,33,18,27,17,24)(14,19,32,40,22,35,21,23)(16,30,36,41,31,26,29,44)(20,37,43,39,38,34,28,42);(5,7,8,12)(6,10,9,11)(13,16,25,36,18,31,17,29)(14,20,32,43,22,38,21,28)(15,26,33,44,27,30,24,41)(19,34,40,42,35,37,23,39);(1,3,4,2)(15,27)(19,35)(23,40)(24,33)(26,30)(34,37)(39,42)(41,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=hi,d^2=gi,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7503982);