Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C2×C4×D4 | C23 |
| socle | C22 | C2×#342 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1 | |
869 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,4,7,6,8,5,3)(9,10)(11,17,18,23)(12,20,21,15)(13,16)(14,19)(22,24);(1,3,5,8,6,7,4,2)(9,11)(10,14,16,22)(12,19)(13,18)(15,17,20,23)(21,24);(11,18)(14,22)(17,23)(19,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=i,e^2=h,f^2=h,b^a=bd,d^a=dg,f^a=fi,g^a=gi,c^b=ci,e^b=egi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72278529, #72278552, #72278640, … |
| maximal normal subgroup of | #72278529, #72278552, #72278640, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34271);