Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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×#3062 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2188 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)(7,9)(8,10)(11,12)(14,19)(15,21)(16,20)(17,26)(22,23)(24,28);(5,7)(6,9)(8,11)(10,12)(13,14,17,19,18,22,26,23)(15,16,21,25,24,20,28,27);(2,4)(7,11)(9,12)(13,15,17,21,18,24,26,28)(14,20,19,27,22,16,23,25);(6,10)(9,12)(13,16)(14,21)(15,23)(17,25)(18,20)(19,24)(22,28)(26,27);(14,22)(16,20)(19,23)(25,27)Polycyclic presentation:
2^9|b^2=f,c^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340483);