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 | |
3512 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)(7,11)(8,18)(9,22)(10,14)(12,21)(13,24)(15,29)(16,28)(17,27)(19,34)(20,35)(23,26)(25,32)(30,36)(31,33);(1,2)(3,4)(5,7,10,17)(6,11,14,27)(8,15,20,31)(9,23,24,16)(12,25,19,30)(13,28,22,26)(18,33,35,29)(21,36,34,32);(2,4)(5,8)(6,12,14,19)(7,15)(9,21)(10,20)(11,25,27,30)(13,18,22,35)(16,32)(17,31)(23,36)(24,34)(26,29,28,33);(5,6)(7,11)(8,19)(9,13)(10,14)(12,20)(15,30)(16,26)(17,27)(18,34)(21,35)(22,24)(23,28)(25,31)(29,36)(32,33)Polycyclic presentation:
2^9|b^2=i,c^2=h,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 | #72306876, #72310726, #72310907, … |
| maximal normal subgroup of | #72306876, #72310726, #72310907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 289641);