Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3506 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3187 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)(13,14,19,28,18,22,17,23)(15,24,25,20,21,27,26,16);(5,6,8,9)(7,10,11,12)(13,15,18,21)(14,20,22,16)(17,25,19,26)(23,27,28,24);(2,4)(6,9)(7,11)(13,16,18,20)(14,21,22,15)(17,27,19,24)(23,25,28,26);(5,7)(6,10)(8,11)(9,12)(14,22)(16,20)(23,28)(24,27);(6,9)(7,11)(13,17,18,19)(14,23,22,28)(15,25,21,26)(16,24,20,27)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,c^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503339);