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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3334 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,20,24,28)(16,25,27,21)(17,22,23,26);(2,4)(13,15,18,24)(14,20,19,28)(16,26,27,22)(17,21,23,25);(6,9)(10,12)(13,16)(14,21)(15,23)(17,24)(18,27)(19,25)(20,22)(26,28);(5,7,8,11)(6,10,9,12)(13,17,18,23)(14,22,19,26)(15,16,24,27)(20,25,28,21);(15,24)(16,27)(20,28)(21,25)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355829);