Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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 | #35236 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1678 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,7,8,5)(17,18)(19,27)(20,26)(21,25)(22,24)(23,28)(29,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,31)(25,27)(26,32);(3,8)(5,7)(10,13)(14,16)(18,24)(20,22)(23,31)(25,26)(27,32)(29,30);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(18,25)(19,28)(24,26)(29,30);(10,13)(14,16)(17,20,21,22)(18,26,25,24)(19,29,28,30)(23,31,27,32)Polycyclic presentation:
2^9|a^2=fi,d^2=fhi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,e^c=ei,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72257444, #72257546, #72264293, … |
| maximal normal subgroup of | #72257444, #72257546, #72264293, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6678611);