Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(14,18)(19,21)(20,27)(22,28);(2,4)(13,14)(15,22)(16,18)(17,19)(20,24)(21,25)(23,28)(26,27);(6,9)(10,12)(14,19)(17,25)(18,21)(20,22)(24,26)(27,28);(13,15,16,23)(14,20,18,27)(17,26,25,24)(19,22,21,28);(5,7,8,11)(6,10,9,12)(14,18)(19,21)(20,27)(22,28)Polycyclic presentation:
2^9|a^2=fi,d^2=i,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72308980, #72308988 |
| maximal normal subgroup of | #72308980, #72308988 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361398);