Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2540 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1983 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14,19,24)(15,20,28,37)(16,21,18,23)(17,30,32,22)(25,34,26,35)(27,42,43,36)(29,40,33,38)(31,44,41,39);(5,6)(7,10)(8,9)(11,12)(13,15,19,28)(14,20,24,37)(16,26,18,25)(17,31,32,41)(21,35,23,34)(22,39,30,44)(27,29,43,33)(36,38,42,40);(1,2,3,4)(6,9)(10,11)(15,25)(16,18)(20,34)(21,23)(26,28)(27,41)(29,33)(31,43)(35,37)(36,44)(38,40)(39,42);(5,7,8,12)(6,10,9,11)(13,16,19,18)(14,21,24,23)(15,26,28,25)(17,29,32,33)(20,35,37,34)(22,38,30,40)(27,31,43,41)(36,39,42,44);(13,17,19,32)(14,22,24,30)(15,27,28,43)(16,29,18,33)(20,36,37,42)(21,38,23,40)(25,41,26,31)(34,44,35,39)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591977);