Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| 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 | #35522 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1904 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)(14,15)(17,18)(20,23)(21,22)(25,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14,17,15,19,22,18,21)(16,20,25,23,24,27,26,28);(2,4)(6,9)(7,11)(13,15,17,22,19,21,18,14)(16,23,25,27,24,28,26,20);(5,7,8,11)(6,10,9,12)(13,16,19,24)(14,20,22,27)(15,23,21,28)(17,25,18,26);(6,9)(10,12)(16,24)(20,27)(23,28)(25,26)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421736);