Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | #35000 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1695 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,6)(7,8)(9,10)(11,14)(12,19,20,15)(13,16)(17,21,23,24)(18,22);(1,2,6,8,4,7,3,5)(9,11)(10,14)(12,17)(13,18)(15,21)(16,22)(19,24)(20,23);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,23)(22,24);(10,16)(14,22)(15,19)(21,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,f^b=fi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72257409, #72257452, #72257496, … |
| maximal normal subgroup of | #72257409, #72257452, #72257496, … |
| maximal quotient of | #72264097 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108601);