Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3052 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)(5,6,8,9)(7,10,11,12)(13,14)(15,24)(16,21)(17,28)(18,23)(19,22)(20,26)(25,27);(1,2,3,4)(13,14)(15,25)(16,20)(17,22)(18,23)(19,28)(21,26)(24,27);(6,9)(10,12)(13,15)(14,20)(16,17)(18,26)(19,27)(21,22)(23,24)(25,28);(5,7,8,11)(6,10,9,12)(13,16,18,27)(14,21,23,25)(15,17,26,19)(20,22,24,28);(13,17,18,19)(14,22,23,28)(15,16,26,27)(20,21,24,25)Polycyclic presentation:
2^9|a^2=gi,b^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6357175);