Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #988785 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4188 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)(6,7)(8,12)(13,14,16,17)(15,19,20,18)(22,26)(23,24);(5,6,9,7)(8,10,12,11)(13,14)(15,19)(16,17)(18,20)(21,22)(23,27)(24,28)(25,26);(8,12)(10,11)(14,17)(18,19)(21,23,25,24)(22,27,26,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,16,20)(14,18,17,19)(21,24,25,23)(22,27,26,28);(5,8)(6,10)(7,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,28)(24,27)(25,26)Polycyclic presentation:
2^9|a^2=fhi,b^2=fg,c^2=gh,d^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.
| maximal subgroup of | #72331959, #72332003, #72332005, … |
| maximal normal subgroup of | #72331959, #72332003, #72332005, … |
| maximal quotient of | #72332532 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 914754);