Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| 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 | |
2043 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)(5,6,8,10)(7,9,11,12)(14,19)(15,23)(17,25)(20,26)(21,24)(22,28);(5,7)(6,9)(8,11)(10,12)(13,14,17,19,18,22,25,28)(15,24,16,26,23,20,27,21);(2,4)(7,11)(9,12)(13,15)(14,20)(16,17)(18,23)(19,21)(22,24)(25,27)(26,28);(5,7)(6,9)(8,11)(10,12)(13,16)(14,21)(15,25)(17,23)(18,27)(19,24)(20,28)(22,26);(7,11)(9,12)(15,23)(16,27)(20,24)(21,26)Polycyclic presentation:
2^9|a^2=hi,b^2=f,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6499038);