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 | #37280 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1793 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14)(15,20)(16,24)(17,22)(18,28)(19,23)(21,25)(26,27);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,18)(14,20,23,28)(16,25,26,22)(17,24,21,27);(6,9)(7,11)(13,16,15,22,19,26,18,25)(14,21,28,27,23,17,20,24);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,24)(16,20)(18,27)(19,21)(23,25)(26,28);(6,9)(10,12)(13,15,19,18)(14,20,23,28)(16,22,26,25)(17,27,21,24)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,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, 6427381);