Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573523 |
| 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 | #36085 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2677 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)(7,12)(8,10)(9,11)(13,14)(15,20)(16,18)(17,21)(19,22)(23,27)(24,36)(25,38)(26,28)(29,33)(30,35)(31,32)(34,37)(39,42)(40,43)(41,44);(5,7,9,8)(6,10,11,12)(13,15,18,26)(14,20,16,28)(17,24,33,40)(19,25,35,27)(21,36,29,43)(22,38,30,23)(31,44,37,39)(32,41,34,42);(2,4)(13,16,18,14)(15,23,26,38)(17,31,33,37)(19,22,35,30)(20,25,28,27)(21,34,29,32)(24,36,40,43)(39,42,44,41);(5,8,9,7)(6,10,11,12)(13,17)(14,21)(15,24)(16,29)(18,33)(19,32)(20,36)(22,31)(23,39)(25,41)(26,40)(27,42)(28,43)(30,37)(34,35)(38,44);(14,16)(15,25)(17,32)(19,35)(20,23)(21,37)(26,27)(28,38)(29,31)(33,34)(36,43)(41,42)Polycyclic presentation:
2^9|b^2=fi,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483718);