Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2021 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)(7,11)(8,10)(9,12)(13,14)(15,21)(16,22)(17,23)(18,35)(19,25)(20,26)(24,28)(27,36)(29,37)(30,38)(31,39)(32,43)(33,40)(34,41)(42,44);(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,30)(17,28)(18,19)(20,29)(22,38)(23,24)(25,35)(26,37)(27,33)(31,42)(32,34)(36,40)(39,44)(41,43);(2,4)(7,12)(9,11)(13,16,19,33,20,34,17,31)(14,22,25,40,26,41,23,39)(15,27,28,30,29,42,18,32)(21,36,24,38,37,44,35,43);(6,10)(9,11)(13,17,20,19)(14,23,26,25)(15,28,29,18)(16,31,34,33)(21,24,37,35)(22,39,41,40)(27,30,42,32)(36,38,44,43);(13,18,17,29,20,28,19,15)(14,24,23,21,26,35,25,37)(16,32,31,42,34,30,33,27)(22,38,39,36,41,43,40,44)Polycyclic presentation:
2^9|b^2=f,c^2=fh,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717694);