Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 | |
2019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,9)(8,10)(11,12)(13,14,20,25)(15,24,28,36)(16,22,31,40)(17,23,19,34)(18,33,27,21)(26,38,42,44)(29,37,32,39)(30,43,41,35);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,18,20,28,19,27)(14,21,34,36,25,33,23,24)(16,26,32,30,31,42,29,41)(22,35,37,44,40,43,39,38);(6,10)(7,11)(13,16,20,31)(14,22,25,40)(15,26,28,42)(17,32,19,29)(18,30,27,41)(21,35,33,43)(23,39,34,37)(24,38,36,44);(1,2,3,4)(6,10)(9,12)(13,17,20,19)(14,23,25,34)(15,27,28,18)(16,29,31,32)(21,36,33,24)(22,37,40,39)(26,30,42,41)(35,38,43,44);(13,18,17,28,20,27,19,15)(14,24,34,21,25,36,23,33)(16,30,32,42,31,41,29,26)(22,38,37,35,40,44,39,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=h,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7518984);