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 | #39462 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2995 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)(13,14,19,27,18,22,17,21)(15,23,26,16,20,28,24,25);(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,26)(18,20)(19,24)(21,28)(22,25)(23,27);(2,4)(6,9)(7,11)(13,16)(14,20)(15,22)(17,23)(18,25)(19,28)(21,26)(24,27);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,23)(18,25)(19,28)(20,22)(21,24)(26,27);(6,9)(7,11)(15,20)(16,25)(23,28)(24,26)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,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, 6503323);