Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23876 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | #23399 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1668 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,11,17,10,29)(7,18,20,13,21,23,30,33)(8,22,24,34,25,14,31,19)(9,26,27,15,28,35,32,36);(1,2)(3,4)(5,7)(6,13)(8,15)(9,19)(10,30)(11,21)(12,20)(14,32)(16,23)(17,33)(18,29)(22,27)(24,35)(25,36)(26,31)(28,34);(2,4)(5,8)(6,14)(7,15)(9,23)(10,31)(11,25)(12,24)(13,32)(16,19)(17,22)(18,28)(20,35)(21,36)(26,30)(27,33)(29,34);(5,9,12,27,11,28,10,32)(6,15,16,35,17,36,29,26)(7,19,20,22,21,34,30,14)(8,23,24,33,25,18,31,13)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,e^c=ei,e^d=ei,f^d=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72320982, #72321557 |
| maximal normal subgroup of | #72320982, #72321557 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 283031);