Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3275 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14)(15,24)(16,29)(17,23)(18,21)(19,22)(20,28)(25,34)(26,35)(27,41)(30,40)(31,44)(32,37)(33,38)(36,43)(39,42);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,26)(17,31)(18,28)(19,25)(21,24)(22,35)(23,39)(27,30)(29,34)(32,42)(33,43)(36,40)(37,44)(38,41);(1,2,3,4)(6,9)(10,11)(14,21)(15,25)(16,19)(20,34)(23,37)(24,35)(26,28)(27,31)(30,33)(36,44)(39,41)(42,43);(5,7,8,12)(6,10,9,11)(13,16,18,19)(14,22,21,29)(15,26,28,25)(17,30,32,33)(20,35,24,34)(23,38,37,40)(27,42,43,31)(36,39,41,44);(13,17,18,32)(14,23,21,37)(15,27,28,43)(16,30,19,33)(20,36,24,41)(22,38,29,40)(25,31,26,42)(34,44,35,39)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6591949);