Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535664 |
| 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 | #37839 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2152 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)(13,14)(15,23)(16,20)(17,21)(18,26)(19,22)(24,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15,17,24,19,25,18,20)(14,16,26,28,22,27,21,23);(2,4)(6,9)(10,12)(13,16,19,27)(14,20,22,24)(15,21,25,26)(17,28,18,23);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,21,22,26)(15,24,25,20)(16,23,27,28);(7,11)(10,12)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452391);