Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #151827 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #12574 |
| commutator | C2×D4 | C2×C42 |
| Frattini, cosocle | C2×#127 | C23 |
| prime radical | #3498 | C22 |
| LCS | G ▷ C2×D4 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C2×#41 ◁ #3498 ◁ G | |
| LECS | G ▷ C2×#127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
581 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,14,7)(4,8)(5,10,19,9)(6,11)(13,27,29,20)(15,26,18,21)(16,24)(17,23,30,25)(22,28,32,31);(1,3)(2,7)(4,13,16,28)(5,18)(6,15)(8,22,24,20)(9,12)(10,26)(11,21)(14,19)(17,31,30,29)(23,27,25,32);(1,4)(2,8)(3,13)(5,17)(6,16)(7,20)(9,25)(10,23)(11,24)(12,27)(14,29)(15,28)(18,31)(19,30)(21,22)(26,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,g^2=i,b^a=bd,d^a=dg,g^a=gh,h^a=hi,c^b=ch,e^b=egi,g^b=gi,h^b=hi,d^c=di,f^c=fi,e^d=ehi,g^d=gi,g^e=gi50 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 59023);