Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213788 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6)(7,15)(8,12)(9,22)(10,14)(11,18)(13,23)(16,30)(17,29)(19,33)(20,35)(21,28)(24,31)(25,32)(26,34)(27,36);(1,2)(3,4)(9,23)(13,22)(15,29)(18,32)(20,36)(27,35)(30,33)(31,34);(2,4)(5,7)(6,11,14,25)(8,19,21,16)(9,18)(10,17)(12,26)(13,15,22,29)(20,34,36,31)(23,32)(24,28)(27,33)(30,35);(5,8,10,21)(6,12,14,28)(7,16,17,19)(9,20,23,36)(11,24,25,26)(13,27,22,35)(15,30,29,33)(18,31,32,34)Polycyclic presentation:
2^9|c^2=h,d^2=i,f^2=i,b^a=be,c^a=cf,f^a=fi,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306335, #72306336, #72306339, … |
| maximal normal subgroup of | #72306335, #72306336, #72306339, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289640);