Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#380 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2658 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)(7,16)(8,21,23,28)(9,24)(10,14)(11,15)(12,19)(13,22,29,25)(17,26,32,36)(18,31)(20,27)(30,33,35,34);(1,2)(3,4)(5,7,9,19,11,18,10,20)(6,12,14,16,15,27,24,31)(8,17,22,33,23,32,25,34)(13,26,28,30,29,36,21,35);(2,4)(5,8)(6,13)(7,17)(9,22)(10,25)(11,23)(12,26)(14,28)(15,29)(16,30)(18,32)(19,33)(20,34)(21,24)(27,36)(31,35);(7,18)(12,27)(16,31)(17,32)(19,20)(26,36)(30,35)(33,34)Polycyclic presentation:
2^9|a^2=h,b^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,d^b=di,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72255617, #72255621, #72255635, … |
| maximal normal subgroup of | #72255617, #72255621, #72255635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 264883);