Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | #16249 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2940 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)(4,8)(5,6)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,20,24,22)(26,30)(27,31);(1,3,5,8)(2,4,6,7)(9,10,12,14)(11,13,15,16)(17,19,21,24)(18,22,23,20)(25,26)(27,28)(29,30)(31,32);(2,6)(3,8)(11,15)(13,16)(18,23)(20,22)(26,30)(27,31);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,20,21,22)(18,19,23,24)(25,27,29,31)(26,28,30,32);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,28)(26,31)(27,30)(29,32)Polycyclic presentation:
2^9|a^2=hi,b^2=fgi,d^2=gh,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 | #72266255, #72266257, #72266258, … |
| maximal normal subgroup of | #72266255, #72266257, #72266258, … |
| maximal quotient of | #72317900 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1853273);