Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(5,6,8,7)(9,10,15,18)(11,14,19,23)(12,20,13,17)(16,21,24,22);(1,2)(3,4)(9,11,14,18,15,19,23,10)(12,20,21,16,13,17,22,24);(9,12,15,13)(10,16,18,24)(11,17,19,20)(14,21,23,22);(2,4)(5,7,8,6)(9,13,15,12)(10,17,18,20)(11,16,19,24)(14,21,23,22)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=ci,d^b=de,f^c=fi,e^d=ei,h^d=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | C4×#22689 |
| maximal normal subgroup of | C4×#22689 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 263696);