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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2226 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)(13,14,19,24)(15,20,28,37)(16,21,31,40)(17,22,18,23)(25,34,29,38)(26,35,43,44)(27,36,33,42)(30,39,32,41);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15,19,28)(14,20,24,37)(16,29,31,25)(17,27,18,33)(21,38,40,34)(22,36,23,42)(26,32,43,30)(35,41,44,39);(2,4)(13,16)(14,21)(15,25)(17,32)(18,30)(19,31)(20,34)(22,41)(23,39)(24,40)(26,27)(28,29)(33,43)(35,36)(37,38)(42,44);(2,4)(5,7,9,8)(6,11,12,10)(13,16)(14,21)(15,26)(17,30)(18,32)(19,31)(20,35)(22,39)(23,41)(24,40)(25,27)(28,43)(29,33)(34,36)(37,44)(38,42);(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,22,24,23)(15,27,28,33)(16,30,31,32)(20,36,37,42)(21,39,40,41)(25,43,29,26)(34,44,38,35)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dh,h^b=hi,e^c=ei,h^d=hi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429620);