Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437415 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2414 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)(7,12)(8,10)(9,11)(13,14)(15,23)(16,20)(17,29)(18,21)(19,25)(22,31)(24,37)(26,36)(27,34)(28,41)(30,33)(32,39)(35,40)(38,44)(42,43);(7,8)(10,12)(13,15,18,26)(14,19,21,33)(16,24,27,38)(17,30,31,25)(20,32,34,43)(22,36,29,23)(28,42,40,39)(35,44,41,37);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,16)(14,20)(15,24)(17,28)(18,27)(19,32)(21,34)(22,35)(23,37)(25,39)(26,38)(29,41)(30,42)(31,40)(33,43)(36,44);(5,8,9,7)(6,10,11,12)(16,27)(20,34)(24,38)(28,40)(32,43)(35,41)(37,44)(39,42);(14,21)(19,33)(20,34)(22,29)(23,36)(32,43)(35,41)(37,44)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370053);