Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35346 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #35565 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,17)(15,19,23,22)(16,20,25,26)(21,27,24,28);(1,2,3,4)(13,15)(14,19)(16,24)(17,22)(18,23)(20,28)(21,25)(26,27);(6,9)(10,12)(13,16,18,25)(14,20,17,26)(15,21,23,24)(19,27,22,28);(13,17,18,14)(15,22,23,19)(16,20,25,26)(21,27,24,28);(15,23)(19,22)(21,24)(27,28);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=i,d^2=i,f^2=h,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=di128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7536315);