Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #763332 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(6,7)(17,18,23,27)(19,25,20,24)(21,32,26,22)(28,31,30,29);(10,13)(14,16)(17,19,23,20)(18,24,27,25)(21,30,26,28)(22,29,32,31);(5,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,20,23,19)(18,25,27,24)(21,30,26,28)(22,29,32,31);(1,2,3,4)(5,7,8,6)(17,21,23,26)(18,22,27,32)(19,28,20,30)(24,31,25,29);(9,11,12,15)(10,14,13,16)(17,22)(18,26)(19,29)(20,31)(21,27)(23,32)(24,28)(25,30)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=i,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=df,e^c=ef92 conjugacy classes — view the full character table.
| maximal subgroup of | #72280699 |
| maximal normal subgroup of | #72280699 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 677482);