Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514020 |
| 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 | C2×#548 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,12)(8,10)(9,11)(13,14,17,22)(15,24,27,19)(16,29,31,38)(18,21,33,23)(20,34,39,32)(25,42,44,37)(26,40,28,36)(30,41,43,35);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,30)(17,27)(18,26)(20,37)(21,36)(22,24)(23,40)(25,32)(28,33)(29,35)(31,43)(34,44)(38,41)(39,42);(13,16,17,31)(14,20,22,39)(15,25,27,44)(18,34,33,32)(19,35,24,41)(21,29,23,38)(26,30,28,43)(36,42,40,37);(2,4)(5,8,9,7)(6,10,11,12)(14,21)(15,26)(16,31)(18,33)(20,38)(22,23)(25,43)(27,28)(29,39)(30,44)(35,41)(36,40);(6,11)(10,12)(14,22)(16,31)(19,24)(21,23)(25,44)(30,43)(32,34)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,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=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425198);