Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2228 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)(7,9)(8,10)(11,12)(13,14,18,25,20,26,19,24)(15,27,28,16,22,17,23,21);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(13,15)(14,21)(16,26)(17,25)(18,23)(19,28)(20,22)(24,27);(2,4)(6,10)(9,12)(13,16,18,17,20,21,19,27)(14,22,25,23,26,15,24,28);(2,4)(13,17,18,21,20,27,19,16)(14,23,25,15,26,28,24,22);(13,18,20,19)(14,24,26,25)(15,23,22,28)(16,17,21,27)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=f,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6348947);