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×#3499 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3410 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)(7,12)(8,19)(9,14)(10,25)(11,16)(13,22)(15,27)(17,31)(18,24)(20,32)(21,34)(23,36)(26,28)(29,35)(30,33);(1,2)(3,4)(5,7,11,14)(6,12,16,9)(8,17,21,30)(10,26,27,18)(13,20,29,23)(15,24,25,28)(19,33,34,31)(22,36,35,32);(2,4)(5,8)(6,13,16,29)(7,17)(9,23,12,20)(10,22)(11,21)(14,30)(15,19,25,34)(18,32)(24,33,28,31)(26,36)(27,35);(5,9,11,12)(6,14,16,7)(8,20,21,23)(10,24,27,28)(13,17,29,30)(15,26,25,18)(19,32,34,36)(22,31,35,33)Polycyclic presentation:
2^9|b^2=i,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 | #72306865, #72306876, #72310726, … |
| maximal normal subgroup of | #72306865, #72306876, #72310726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289642);