Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437758 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
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×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3788 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,20,17)(15,23,27,24)(16,18,22,19)(21,28,25,26);(1,2)(3,4)(13,15,19,26,20,27,18,28)(14,21,22,23,17,25,16,24);(2,4)(6,9)(10,12)(13,16)(14,19)(15,24)(17,18)(20,22)(21,26)(23,27)(25,28);(2,4)(5,7,8,11)(6,10,9,12)(13,17,20,14)(15,25,27,21)(16,18,22,19)(23,26,24,28);(13,18,20,19)(14,22,17,16)(15,26,27,28)(21,24,25,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,e^b=ei,e^c=ei,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356368);