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 | |
2178 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)(13,14,19,24)(15,20,28,36)(16,21,30,38)(17,22,18,23)(25,33,42,44)(26,34,41,43)(27,35,32,40)(29,37,31,39);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,25)(17,27)(18,32)(19,28)(21,33)(22,35)(23,40)(24,36)(26,29)(30,42)(31,41)(34,37)(38,44)(39,43);(2,4)(13,16)(14,21)(15,25)(17,31)(18,29)(19,30)(20,33)(22,39)(23,37)(24,38)(26,32)(27,41)(28,42)(34,40)(35,43)(36,44);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,26)(17,29)(18,31)(19,30)(20,34)(22,37)(23,39)(24,38)(25,32)(27,42)(28,41)(33,40)(35,44)(36,43);(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,22,24,23)(15,27,28,32)(16,29,30,31)(20,35,36,40)(21,37,38,39)(25,26,42,41)(33,34,44,43)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,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, 7429580);