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 |
| prime radical | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2833 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,16)(11,13,12,15)(17,18,21,22)(19,23,24,20)(25,26)(27,28)(29,30)(31,32);(2,5)(6,7)(9,11)(10,15)(12,14)(13,16)(17,19,21,24)(18,20,22,23)(25,26,29,30)(27,32,31,28);(3,8)(6,7)(10,16)(11,12)(18,22)(20,23)(26,30)(27,31);(1,3,4,8)(2,6,5,7)(9,12,14,11)(10,13,16,15)(17,20)(18,19)(21,23)(22,24)(25,27,29,31)(26,28,30,32);(1,2)(3,6)(4,5)(7,8)(9,13)(10,11)(12,16)(14,15)(17,19)(18,20)(21,24)(22,23)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|a^2=fhi,b^2=fi,d^2=fgh,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 | #72326579, #72326596, #72326641, … |
| maximal normal subgroup of | #72326579, #72326596, #72326641, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1910633);