Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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×#3275 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2251 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(14,19)(15,22)(16,18)(20,27)(23,28)(24,26);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,14,17,19)(15,23,24,20)(16,25,18,21)(22,27,26,28);(7,12)(9,10)(13,15)(14,20)(16,22)(17,24)(18,26)(19,23)(21,27)(25,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,23)(16,25)(17,19)(18,21)(20,24)(22,27)(26,28);(7,12)(9,10)(13,16,17,18)(14,21,19,25)(15,22,24,26)(20,27,23,28)Polycyclic presentation:
2^9|b^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504169);