Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #196993 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #60558 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5,6,7)(9,10,13,16,14,18,12,17)(11,19,22,23,20,24,21,15);(1,2,3,7,4,6,8,5)(9,11)(10,15,18,23)(12,21)(13,22)(14,20)(16,19,17,24);(10,16)(12,13)(15,19)(17,18)(21,22)(23,24);(11,20)(15,23)(19,24)(21,22)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,f^b=fh,g^b=gi,e^c=eh,f^c=fh,g^c=gh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239090, #72239111, #72239112, … |
| maximal normal subgroup of | #72239090, #72239111, #72239112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104249);