Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535655 |
| 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 | C2×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2065 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)(15,16)(17,20)(18,23)(19,21)(22,24)(25,27)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15,16,14,19,23,18,21)(17,22,24,20,26,27,25,28);(2,4)(6,9)(10,12)(13,16,19,18)(14,15,21,23)(17,25,26,24)(20,27,28,22);(5,7)(6,10)(8,11)(9,12)(13,17,19,26)(14,20,21,28)(15,22,23,27)(16,24,18,25);(7,11)(10,12)(17,26)(20,28)(22,27)(24,25)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=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6452375);