Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1771 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)(7,11)(8,10)(9,12)(13,14)(15,26)(16,21)(17,25)(18,23)(19,24)(20,28)(22,30)(27,41)(29,36)(31,35)(32,40)(33,39)(34,43)(37,42)(38,44);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,17)(18,31)(19,28)(21,22)(23,38)(24,26)(25,36)(27,42)(29,30)(32,43)(33,44)(34,37)(35,39)(40,41);(1,2,3,4)(6,10)(7,12)(13,16,30,28,19,29,17,15)(14,21,22,20,24,36,25,26)(18,31,42,43,33,44,32,27)(23,35,37,34,39,38,40,41);(6,10)(9,11)(13,17,19,30)(14,22,24,25)(15,16,28,29)(18,32,33,42)(20,21,26,36)(23,37,39,40)(27,31,43,44)(34,35,41,38);(13,18,17,32,19,33,30,42)(14,23,25,40,24,39,22,37)(15,27,29,44,28,43,16,31)(20,34,21,35,26,41,36,38)Polycyclic presentation:
2^9|b^2=fi,c^2=ghi,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723399);