Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514000 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1978 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14)(15,21)(16,27)(17,22)(18,23)(19,24)(20,25)(26,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(15,25)(16,26)(17,18)(20,21)(22,23)(27,28);(5,8,9,7)(6,10,11,12)(13,15,19,26)(14,20,24,27)(16,18,25,17)(21,23,28,22);(2,4)(13,16,19,25)(14,21,24,28)(15,17,26,18)(20,22,27,23);(6,11)(10,12)(13,17,19,18)(14,22,24,23)(15,16,26,25)(20,21,27,28)Polycyclic presentation:
2^9|b^2=f,c^2=fi,d^2=i,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7423063);