Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518952 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2086 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)(11,13)(12,14)(15,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,30)(28,31)(29,32);(1,3,5,4)(2,6,7,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,23)(20,22)(21,27)(24,25)(26,30)(28,29)(31,32);(10,14)(11,15)(18,24)(19,20)(26,32)(27,28);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(17,20,29,28,22,19,21,27)(18,25,32,31,24,23,26,30);(2,7)(6,8)(17,21,22,29)(18,26,24,32)(19,27,20,28)(23,30,25,31)Polycyclic presentation:
2^9|b^2=f,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72283106, #72283754, #72283999 |
| maximal normal subgroup of | #72283106, #72283754, #72283999 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7516988);