Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 | #25162 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1885 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)(5,6,8,9)(7,11,12,10)(13,14,17,16)(15,21,24,19)(18,20,26,25)(22,23,28,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,24)(14,19,16,21)(18,27,26,23)(20,28,25,22);(2,4)(6,9)(10,11)(13,16,17,14)(15,22,24,28)(18,20,26,25)(19,27,21,23);(5,7,8,12)(6,10,9,11)(14,16)(19,21)(20,25)(22,28);(15,23)(18,26)(19,22)(20,25)(21,28)(24,27)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307915 |
| maximal normal subgroup of | #72307915 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578014);