Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #612424 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2906 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(14,17)(15,19)(21,22)(23,27)(24,25)(26,28);(5,6)(7,9)(8,10)(11,12)(13,14,18,17)(15,16,19,20)(22,25)(26,27);(13,15)(14,16)(17,20)(18,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,16,18,20)(14,19,17,15);(5,7,8,12)(6,9,10,11)(13,17,18,14)(15,20,19,16)(21,23,24,28)(22,26,25,27)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 519893);