Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3508 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)(7,12)(8,10)(9,11)(13,14,19,17)(15,23,27,28)(16,22,18,21)(20,24,25,26);(1,2)(3,4)(13,15,18,26,19,27,16,24)(14,20,22,23,17,25,21,28);(2,4)(5,7,9,8)(6,10,11,12)(13,16,19,18)(14,21,17,22)(15,24,27,26)(20,28,25,23);(2,4)(5,8,9,7)(6,10,11,12)(13,17,19,14)(15,25,27,20)(16,22,18,21)(23,26,28,24);(13,16,19,18)(14,22,17,21)(15,26,27,24)(20,28,25,23)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,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^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359295);