Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10,13,12)(11,14,15,16)(17,18)(19,23)(20,22)(21,24)(26,27)(28,32);(2,3)(4,8)(10,12)(14,16)(17,18,20,22)(19,21,24,23)(26,27)(30,31);(2,3)(6,7)(11,15)(14,16)(19,24)(21,23)(25,26,29,27)(28,30,32,31);(1,3,5,2)(4,7,8,6)(9,11,13,15)(10,14,12,16)(17,19)(18,21)(20,24)(22,23)(25,27,29,26)(28,31,32,30);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,16,15,14)(17,18)(19,21)(20,22)(23,24)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=fhi,b^2=hi,c^2=h,d^2=fi,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72303510, #72303512, #72303515, … |
| maximal normal subgroup of | #72303510, #72303512, #72303515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1923986);