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×#3012 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2850 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)(7,12)(8,10)(9,11)(13,14)(15,19)(16,21)(17,26)(18,22)(20,24)(23,27)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,23)(14,19,22,27)(16,25,17,24)(20,21,28,26);(2,4)(13,15)(14,20)(16,24)(17,25)(18,23)(19,26)(21,27)(22,28);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(15,23)(19,27)(20,28)(24,25);(2,4)(13,16,18,17)(14,21,22,26)(15,24,23,25)(19,20,27,28)Polycyclic presentation:
2^9|b^2=fi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=ehi77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6484199);