Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2502 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,10,14,11,16,9,15)(7,17,20,12,19,29,21,28)(8,22,24,32,25,31,26,13)(18,30,35,36,23,27,34,33);(5,7)(6,12)(8,23)(9,21)(10,20)(11,19)(13,30)(14,29)(15,17)(16,28)(18,25)(22,36)(24,34)(26,35)(27,32)(31,33);(2,4)(5,8,9,24,11,25,10,26)(6,13,14,31,16,32,15,22)(7,18,20,34,19,23,21,35)(12,27,17,33,28,30,29,36);(7,19)(12,28)(17,29)(18,23)(20,21)(27,30)(33,36)(34,35)Polycyclic presentation:
2^9|a^2=h,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,e^b=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265107);