Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| 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 | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3273 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,8)(7,11,12,10)(13,14)(15,20)(16,19)(17,18)(21,22,25,24)(23,27,28,26);(2,4)(6,8)(10,11)(14,18)(16,20)(21,23)(22,26)(24,27)(25,28);(1,2,3,4)(7,12)(10,11)(14,18)(15,19)(23,28)(26,27);(5,7)(6,10)(8,11)(9,12)(13,15,17,19)(14,16,18,20)(21,23)(22,27)(24,26)(25,28);(5,8,9,6)(7,11,12,10)(13,16)(14,19)(15,18)(17,20)(21,24,25,22)(23,26,28,27)Polycyclic presentation:
2^9|a^2=fg,c^2=fhi,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72336799, #72336878, #72336913 |
| maximal normal subgroup of | #72336799, #72336878, #72336913 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 612696);