Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62842 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
11102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,4,3)(6,8)(9,10)(11,16)(12,17)(13,21)(14,15)(18,22)(19,20)(23,24);(5,6)(7,8)(10,15)(12,16)(18,21)(20,23);(1,3,4,2)(10,15)(11,17)(18,21)(19,24);(9,11,14,17)(10,16,15,12)(13,19,22,24)(18,23,21,20);(9,12,14,16)(10,17,15,11)(13,20,22,23)(18,24,21,19);(9,13)(10,18)(11,19)(12,20)(14,22)(15,21)(16,23)(17,24)Polycyclic presentation:
2^9|a^2=i,c^2=i,d^2=h,e^2=h,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532922);