Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470161 |
| 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 | #37616 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2285 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,23)(16,22)(17,20)(18,21)(19,25)(24,27)(26,28);(1,2,3,4)(7,8)(10,11)(13,14)(15,24)(16,20)(17,22)(18,21)(19,26)(23,27)(25,28);(5,7,9,8)(6,10,12,11)(13,15)(14,19)(16,25)(17,27)(18,26)(20,23)(21,24)(22,28);(5,8,9,7)(6,11,12,10)(13,16,18,17)(14,20,21,22)(15,25,26,27)(19,23,24,28);(6,12)(10,11)(14,21)(19,24)(20,22)(23,28)Polycyclic presentation:
2^9|b^2=f,c^2=hi,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394917);