Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18376 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6,7,8)(9,10,15,18,14,11,13,17)(12,16,22,24,21,19,20,23);(1,2,3,6,5,7,4,8)(9,11,14,10)(12,19,21,16)(13,17,15,18)(20,23,22,24);(1,3,5,4)(2,6,7,8)(9,12,14,21)(10,16,11,19)(13,22,15,20)(17,24,18,23);(2,7)(6,8)(10,11)(16,19)(17,18)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=dh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,g^c=gi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 102195);