Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23472 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1658 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)(7,11,16,23)(8,12,19,26)(9,14)(10,13)(15,22)(17,24,30,35)(18,25)(20,27,32,33)(21,28)(29,34)(31,36);(5,7)(6,11,13,23)(8,18,19,31)(9,17)(10,16)(12,25)(14,24,28,35)(15,20,29,32)(21,30)(22,33)(26,36)(27,34);(5,8)(6,12,13,26)(7,15,16,29)(9,20)(10,19)(11,22)(14,27,28,33)(17,18,30,31)(21,32)(23,34)(24,25)(35,36);(7,16)(11,23)(15,29)(17,30)(18,31)(22,34)(24,35)(25,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=e,g^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,h^b=hi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265444);