Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| 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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,7,8)(10,13)(14,15)(17,18,20,22)(19,23,24,21)(26,29)(27,31);(2,6)(4,8)(9,10)(11,15)(12,13)(14,16)(17,19,20,24)(18,21,22,23)(25,26)(27,32)(28,29)(30,31);(2,6)(3,7)(10,13)(14,15)(18,22)(21,23)(25,26,28,29)(27,30,31,32);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24)(25,26)(27,30)(28,29)(31,32);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,21)(20,24)(22,23)(25,27)(26,30)(28,31)(29,32)Polycyclic presentation:
2^9|a^2=fh,b^2=hi,c^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232590, #72232605, #72232609, … |
| maximal normal subgroup of | #72232590, #72232605, #72232609, … |
| maximal quotient of | #72328480, #72328774, #72328783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1024722);