Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196248 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3748 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)(8,19,21,12)(9,22)(10,14)(11,17)(13,24)(16,29,32,36)(18,30)(20,27,23,28)(25,31,35,33)(26,34);(5,7,9,17,10,18,24,34)(6,11,13,15,14,26,22,30)(8,16,23,33,21,32,20,31)(12,25,28,29,19,35,27,36);(5,8)(6,12)(7,16)(9,23)(10,21)(11,25)(13,28)(14,19)(15,29)(17,33)(18,32)(20,24)(22,27)(26,35)(30,36)(31,34);(2,4)Polycyclic presentation:
2^9|a^2=h,b^2=eh,e^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,f^b=fi,e^c=ei,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72302363, #72302579, #72302583, … |
| maximal normal subgroup of | #72302363, #72302579, #72302583, … |
| maximal quotient of | #72316795, #72316996, #72317024 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299890);