Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469989 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3135 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(7,11)(10,12)(13,14)(15,25)(16,21)(17,24)(18,22)(19,23)(20,28)(26,41)(27,36)(29,35)(30,44)(31,40)(32,39)(33,38)(34,42)(37,43);(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,23,36)(16,30,32,26)(17,28,18,27)(21,37,39,34)(22,25,24,35)(31,42,33,43)(38,41,40,44);(2,4)(6,9)(10,12)(13,16)(14,21)(15,26)(17,33)(18,31)(19,32)(20,34)(22,40)(23,39)(24,38)(25,41)(27,43)(28,42)(29,30)(35,44)(36,37);(6,9)(10,12)(13,17,19,18)(14,22,23,24)(15,27,29,28)(16,31,32,33)(20,35,36,25)(21,38,39,40)(26,42,30,43)(34,41,37,44);(5,7,8,11)(6,10,9,12)(14,23)(16,32)(20,36)(22,24)(25,35)(26,30)(31,33)(42,43)Polycyclic presentation:
2^9|b^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398097);