Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2897 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,22)(16,20)(17,18)(19,21)(23,26)(24,27)(25,28);(1,2,3,4)(13,15,22,14,19,17,18,21)(16,20,27,23,25,28,26,24);(5,7,8,12)(6,9,10,11)(13,16)(14,20)(15,23)(17,24)(18,27)(19,25)(21,28)(22,26);(13,17,19,15)(14,18,21,22)(16,24,25,23)(20,27,28,26);(16,25)(20,28)(23,24)(26,27)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=g,d^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352965);