Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682420 |
| 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 | #36360 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1706 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)(4,6)(5,8)(17,18)(19,26)(20,24)(21,30)(22,25)(23,27)(28,32)(29,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16);(2,6)(3,8)(10,13)(14,16)(17,19,21,27,22,29,20,28)(18,23,24,26,25,32,30,31);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,32,26);(10,13)(14,16)(17,20,22,21)(18,24,25,30)(19,28,29,27)(23,26,32,31)Polycyclic presentation:
2^9|c^2=fg,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72256838, #72256850, #72256971 |
| maximal normal subgroup of | #72256838, #72256850, #72256971 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6693315);