Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499814 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3318 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)(13,14)(15,20)(16,22)(17,19)(18,21)(23,28)(24,26)(25,27);(5,6)(7,10)(8,9)(11,12)(14,19)(15,23)(16,17)(20,27)(21,22)(26,28);(2,4)(6,9)(7,11)(13,15,18,23)(14,20,21,28)(16,25,17,24)(19,26,22,27);(5,7,8,11)(6,10,9,12)(14,21)(19,22)(20,28)(26,27);(6,9)(10,12)(13,16,18,17)(14,22,21,19)(15,24,23,25)(20,26,28,27)Polycyclic presentation:
2^9|c^2=i,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6419872);