Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32700 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1518 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)(7,15,18,11)(8,19)(9,23,24,13)(10,14)(12,22)(16,27,20,25)(17,29)(21,28,34,32)(26,31)(30,36)(33,35);(5,7)(6,11)(8,20)(9,17)(10,18)(12,27)(13,29)(14,15)(16,22)(19,25)(21,30)(23,26)(24,31)(28,36)(32,35)(33,34);(2,4)(5,8)(6,12,14,19)(7,16,18,20)(9,21)(10,22)(11,25)(13,28,23,32)(15,27)(17,30,31,33)(24,34)(26,35)(29,36);(2,4)(5,9)(6,13)(7,17)(8,21)(10,24)(11,26)(12,28)(14,23)(15,29)(16,30)(18,31)(19,32)(20,33)(22,34)(25,35)(27,36)Polycyclic presentation:
2^9|a^2=h,c^2=egh,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283596, #72283644, #72283734, … |
| maximal normal subgroup of | #72283596, #72283644, #72283734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 297613);