Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #31414 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1009 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)(5,6,11,16,9,15,10,14)(7,17,21,29,19,12,20,28)(8,22,24,31,26,32,25,13)(18,34,36,27,35,30,23,33);(5,7)(6,12)(8,23,26,36)(9,19)(10,20)(11,21)(13,30,31,34)(14,29)(15,17)(16,28)(18,25,35,24)(22,33,32,27);(5,8)(6,13,15,31)(7,18)(9,26)(10,24)(11,25)(12,27,17,33)(14,22,16,32)(19,35)(20,36)(21,23)(28,34,29,30);(6,14)(7,19)(10,11)(12,28)(13,22)(15,16)(17,29)(18,35)(24,25)(27,34)(30,33)(31,32)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=efg,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,g^a=gi,c^b=cg,d^b=de,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72284999, #72285470, #72285613 |
| maximal normal subgroup of | #72284999, #72285470, #72285613 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 293609);