Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514060 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #30412 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1529 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,13)(15,16)(17,18,20,23)(19,22,25,30)(21,24,28,31)(26,32,27,29);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,20,25)(18,22,23,30)(21,26,28,27)(24,32,31,29);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(18,23)(19,25)(24,31)(26,27);(10,13)(14,16)(18,23)(19,26)(21,28)(22,29)(25,27)(30,32);(2,7)(6,8)(9,11,12,15)(10,14,13,16)(18,23)(22,30)(24,31)(29,32)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72313261, #72313531, #72313550, … |
| maximal normal subgroup of | #72313261, #72313531, #72313550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7481897);