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 | #37276 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1836 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)(15,16)(17,20)(18,24)(19,21)(22,23)(25,27)(26,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,18)(14,16,21,24)(17,22,25,26)(20,23,27,28);(6,9)(7,11)(13,16,15,14,19,24,18,21)(17,23,22,20,25,28,26,27);(5,7,8,11)(6,10,9,12)(13,17,19,25)(14,20,21,27)(15,22,18,26)(16,23,24,28);(6,9)(10,12)(17,25)(20,27)(22,26)(23,28)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=gi,d^2=h,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, 6427378);