Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
10946 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)(3,4)(6,8)(9,10,14,16)(11,15,19,24)(12,17,21,23)(13,18,20,22);(5,6)(7,8)(9,11,14,19)(10,15,16,24)(12,20,21,13)(17,22,23,18);(2,4)(10,16)(11,19)(12,21)(18,22);(1,2)(3,4)(9,12)(10,17)(11,20)(13,19)(14,21)(15,22)(16,23)(18,24);(9,13,14,20)(10,18,16,22)(11,21,19,12)(15,23,24,17);(9,10)(11,15)(12,17)(13,18)(14,16)(19,24)(20,22)(21,23)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7590906);