Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×#453 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
799 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,4,3,6,8)(9,10)(11,22)(12,23)(13,18)(14,20)(15,19)(16,21)(17,24);(1,3,5,8,4,2,6,7)(9,11)(10,13)(12,21)(14,23)(15,18)(16,17)(19,24)(20,22);(2,3)(7,8)(9,12)(10,17)(11,21)(13,16)(14,24)(15,22)(18,20)(19,23);(9,13,16,12)(10,18,21,23)(11,20,17,19)(14,24,15,22)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=h,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,c^b=ci,d^b=de,e^b=eh,d^c=dh,f^c=fh,e^d=eh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72215940 |
| maximal normal subgroup of | #72215940 |
| maximal quotient of | #72215904, #72215911 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 134367);