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 | |
2117 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)(15,26)(16,29)(17,32)(18,24)(19,25)(20,23)(21,28)(22,31)(27,42)(30,40)(33,44)(34,38)(35,41)(36,39)(37,43);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,17)(18,34)(19,28)(20,31)(22,32)(23,29)(24,40)(25,26)(27,33)(30,35)(36,43)(37,44)(38,41)(39,42);(2,4)(7,12)(9,11)(13,16,17,15,19,31,20,28)(14,22,32,21,25,29,23,26)(18,30,33,27,35,34,36,43)(24,38,44,37,41,40,39,42);(6,10)(9,11)(13,17,19,20)(14,23,25,32)(15,16,28,31)(18,33,35,36)(21,29,26,22)(24,39,41,44)(27,30,43,34)(37,40,42,38);(13,18,20,36,19,35,17,33)(14,24,23,39,25,41,32,44)(15,27,16,30,28,43,31,34)(21,37,22,38,26,42,29,40)Polycyclic presentation:
2^9|c^2=fh,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, 6717074);