Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7300916 |
| 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 | #31396 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,6,4,8)(9,10,13,12)(11,14,15,16)(17,18,22,27)(19,23,20,25)(21,30,31,26)(24,29,32,28);(1,2)(3,8)(4,6)(5,7)(9,11)(10,14)(12,16)(13,15)(17,19,22,20)(18,23,27,25)(21,28,31,29)(24,26,32,30);(10,12)(11,15)(17,19,22,20)(18,24,27,32)(21,29,31,28)(23,26,25,30);(1,3,5,4)(2,6,7,8)(9,12,13,10)(11,14,15,16)(17,20,22,19)(18,25,27,23)(21,29,31,28)(24,30,32,26);(1,4,5,3)(2,6,7,8)(11,15)(14,16)(19,20)(23,25)(24,32)(28,29)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=i,d^2=fhi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7317002);