Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13398 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #481 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | #1874 |
| prime radical | C2×#3054 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×M16 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1088 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6)(7,15,17,32)(8,19)(9,23)(10,13)(11,16,26,35)(12,22)(14,25)(18,28,24,33)(20,30,36,27)(21,31)(29,34);(5,7)(6,11)(8,20)(9,24)(10,17)(12,28)(13,26)(14,30)(15,31)(16,34)(18,25)(19,32)(21,36)(22,35)(23,27)(29,33);(2,4)(8,21)(11,26)(12,29)(16,35)(19,31)(20,36)(22,34)(27,30)Polycyclic presentation:
2^9|a^2=g,d^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,f^a=fi,h^a=hi,c^b=cf,d^b=dhi,g^b=gi,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280589 |
| maximal normal subgroup of | #72280589 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54823);