Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393906 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| prime radical | C2×#2010 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(8,19)(11,23)(12,25)(15,28)(21,32)(24,34)(27,35)(29,36);(1,2,3,4)(5,6)(7,15,17,28)(8,12)(9,14)(10,13)(11,18,23,22)(16,29,30,36)(19,25)(20,26)(21,27)(24,31,34,33)(32,35);(5,7)(6,11)(8,16)(9,22)(10,17)(12,24)(13,23)(14,28)(15,26)(18,20)(19,30)(21,33)(25,34)(27,36)(29,35)(31,32);(5,8,20,32,10,19,9,21)(6,12,26,35,13,25,14,27)(7,16,18,31,17,30,22,33)(11,24,15,29,23,34,28,36)Polycyclic presentation:
2^9|b^2=e,d^2=gh,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei,g^c=gi,h^c=hi,f^d=fi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321226);