Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3423 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)(6,9)(10,12)(13,14)(15,20)(16,18)(17,19)(21,22,26,28)(23,24,25,27);(5,6,8,9)(7,10,11,12)(13,15,18,17)(14,19,16,20)(21,23,26,25)(22,27,28,24);(7,11)(10,12)(15,17)(19,20)(22,28)(23,25);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,19,17,20)(21,24)(22,23)(25,28)(26,27);(5,7,8,11)(6,10,9,12)(13,17,18,15)(14,19,16,20)(21,25,26,23)(22,24,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=f,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1991691);