Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2773 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)(7,9)(8,10)(11,12)(13,14)(15,19)(16,20)(17,22)(18,21)(23,31)(24,32)(25,33)(26,34)(27,35)(28,36)(29,37)(30,38)(39,42)(40,44)(41,43);(2,4)(5,7,8,11)(6,9,10,12)(15,23)(16,26)(17,28)(19,31)(20,34)(22,36)(25,29)(27,30)(33,37)(35,38)(39,41)(42,43);(6,10)(9,12)(13,15)(14,19)(16,24)(17,29)(18,25)(20,32)(21,33)(22,37)(23,28)(26,39)(27,40)(30,41)(31,36)(34,42)(35,44)(38,43);(13,16,18,27)(14,20,21,35)(15,24,25,40)(17,30,28,26)(19,32,33,44)(22,38,36,34)(23,39,29,41)(31,42,37,43);(14,21)(15,25)(20,35)(22,36)(23,29)(24,40)(34,38)(39,41)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347732);