Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,20)(16,21)(17,23)(18,24)(19,25)(22,32)(26,34)(27,44)(28,36)(29,37)(30,38)(31,39)(33,41)(35,42)(40,43);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,19,30)(14,20,25,38)(16,26,32,43)(17,29,18,28)(21,34,22,40)(23,37,24,36)(27,33,42,31)(35,39,44,41);(2,4)(13,16)(14,21)(15,26)(17,33)(18,31)(19,32)(20,34)(22,25)(23,41)(24,39)(27,28)(29,42)(30,43)(35,37)(36,44)(38,40);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,22)(15,27)(17,31)(18,33)(19,32)(20,35)(21,25)(23,41)(24,39)(26,28)(29,43)(30,42)(34,37)(36,40)(38,44);(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,23,25,24)(15,28,30,29)(16,31,32,33)(20,36,38,37)(21,39,22,41)(26,42,43,27)(34,35,40,44)Polycyclic presentation:
2^9|b^2=fi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429668);