Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23906 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | #60558 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,3,8,7)(9,10,13,17,14,11,15,19)(12,16,21,23,20,18,22,24);(1,3)(2,5)(4,6)(7,8)(9,11,13,19,14,10,15,17)(12,18,21,24,20,16,22,23);(1,2)(3,5)(4,6)(7,8)(9,12)(10,16)(11,18)(13,21)(14,20)(15,22)(17,23)(19,24);(2,3)(6,7)(10,11)(12,20)(17,19)(21,22)Polycyclic presentation:
2^9|a^2=g,b^2=egh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=di,e^b=eh,g^b=gh,d^c=di,f^c=fh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72264953, #72265464, #72265487 |
| maximal normal subgroup of | #72264953, #72265464, #72265487 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100975);