Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2106 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)(15,26,28,20)(16,30,29,21)(17,25,31,22)(18,23,34,38)(27,40,41,35)(32,43,42,36)(33,39,44,37);(5,7)(6,9)(8,12)(10,11)(13,15,19,28)(14,20,24,26)(16,31,29,17)(18,32,34,42)(21,22,30,25)(23,36,38,43)(27,33,41,44)(35,39,40,37);(2,4)(7,12)(9,11)(13,16,17,15,19,29,31,28)(14,21,25,20,24,30,22,26)(18,32,33,27,34,42,44,41)(23,36,39,35,38,43,37,40);(6,10)(9,11)(13,17,19,31)(14,22,24,25)(15,16,28,29)(18,33,34,44)(20,30,26,21)(23,37,38,39)(27,32,41,42)(35,43,40,36);(13,18,17,33,19,34,31,44)(14,23,25,39,24,38,22,37)(15,27,29,42,28,41,16,32)(20,35,30,43,26,40,21,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fhi,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717086);