Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200042 |
| 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 | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1472 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,5)(4,6)(7,8)(9,10)(11,13)(12,15)(14,16)(17,18,22,27)(19,23,29,24)(20,25,30,32)(21,31,28,26);(1,3)(2,5)(4,7)(6,8)(9,11,13,10)(12,15,16,14)(17,19)(18,23)(20,21)(22,29)(24,27)(25,31)(26,32)(28,30);(3,7)(5,8)(10,11)(14,15)(17,20)(18,24)(19,21)(22,30)(23,27)(25,31)(26,32)(28,29);(2,6)(3,7)(9,12,13,16)(10,14,11,15)(17,20)(18,25)(19,28)(21,29)(22,30)(23,26)(24,31)(27,32);(3,7)(5,8)(10,11)(14,15)(19,29)(20,30)(23,24)(25,32)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfhi,e^b=ehi,e^c=ei,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7110202);