Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573485 |
| 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 | #36213 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2875 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)(7,12)(8,10)(9,11)(13,14,19,16)(15,20,25,26)(17,21,18,22)(23,27,28,24);(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,26)(17,24)(18,27)(19,25)(21,23)(22,28);(2,4)(13,16,19,14)(15,23,25,28)(17,21,18,22)(20,27,26,24);(5,8,9,7)(6,10,11,12)(13,15)(14,20)(16,26)(17,24)(18,27)(19,25)(21,23)(22,28);(13,17)(14,21)(16,22)(18,19)(23,28)(24,27)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483772);