Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470884 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2361 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)(7,9)(8,10)(11,12)(13,14)(15,21)(16,28)(17,22)(18,33)(19,24)(20,26)(23,29)(25,35)(27,36)(30,41)(31,37)(32,43)(34,40)(38,42)(39,44);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,26,19,27,29,16)(14,20,23,21,24,28,33,36)(17,25,32,40,31,39,42,30)(22,34,38,35,37,41,43,44);(7,11)(9,12)(13,16,29,27,19,26,18,15)(14,21,33,20,24,36,23,28)(17,30,42,39,31,40,32,25)(22,35,43,34,37,44,38,41);(7,11)(9,12)(13,17,19,31)(14,22,24,37)(15,25,27,39)(16,30,26,40)(18,32,29,42)(20,34,28,41)(21,35,36,44)(23,38,33,43);(6,10)(9,12)(17,31)(22,37)(25,39)(30,40)(32,42)(34,41)(35,44)(38,43)Polycyclic presentation:
2^9|b^2=f,c^2=gi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380609);