Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| 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 | |
2520 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)(6,8)(7,11)(15,19)(18,20)(22,25)(23,27);(5,6,9,8)(7,10,11,12)(14,17)(18,20)(23,27)(26,28);(6,8)(10,12)(13,14,16,17)(15,18,19,20)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,7,9,11)(6,10,8,12)(13,15)(14,18)(16,19)(17,20)(21,23)(22,26)(24,27)(25,28);(5,8,9,6)(7,12,11,10)Polycyclic presentation:
2^9|a^2=ghi,b^2=f,c^2=fgh,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,e^c=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 914745);