Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667604 |
| 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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1775 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)(5,6,8,9)(7,11,12,10)(13,14,18,23)(15,24,28,20)(16,29,19,21)(17,31,33,22)(25,37,26,35)(27,36,43,41)(30,38,34,40)(32,39,42,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,28)(14,20,23,24)(16,26,19,25)(17,32,33,42)(21,35,29,37)(22,39,31,44)(27,30,43,34)(36,40,41,38);(2,4)(6,9)(10,11)(15,25)(16,19)(20,35)(21,29)(24,37)(26,28)(27,42)(30,34)(32,43)(36,44)(38,40)(39,41);(5,7,8,12)(6,10,9,11)(13,16,18,19)(14,21,23,29)(15,26,28,25)(17,30,33,34)(20,35,24,37)(22,38,31,40)(27,32,43,42)(36,44,41,39);(13,17,18,33)(14,22,23,31)(15,27,28,43)(16,30,19,34)(20,36,24,41)(21,38,29,40)(25,42,26,32)(35,44,37,39)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578021);