Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2230 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #476 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,16,18,11)(12,22)(13,23)(14,24)(17,20,21,19);(2,4)(9,10,14,19,15,18,24,20)(11,12,17,22,16,23,21,13);(9,11)(10,12)(13,20)(14,17)(15,16)(18,23)(19,22)(21,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=h,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,g^a=gi,h^a=hi,d^b=dh,e^b=eh,f^b=fi,h^b=hi,d^c=dh,e^c=eh,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72242475, C4×#12823 |
| maximal normal subgroup of | #72242475, C4×#12823 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53934);