Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9938 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #392 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,18)(17,24)(20,23)(21,27)(22,26)(25,28);(5,7)(6,9)(8,11)(10,12)(14,18)(15,21,22,25)(19,27,26,28)(20,24);(9,12)(13,15)(14,19)(16,22)(17,25)(18,26)(20,27)(21,23)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=dh,f^2=h,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,e^b=ei,d^c=dh,f^c=fh68 conjugacy classes — view the full character table.
| maximal subgroup of | #72257806, #72264049, #72264056 |
| maximal normal subgroup of | #72257806, #72264049, #72264056 |
| maximal quotient of | #72264922 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 23534);