Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24404 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C4×#120 |
| prime radical | #23595 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1181 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,8)(9,10,13,12,14,16,22,21)(11,15,19,20,17,23,24,18);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,20)(13,19)(14,17)(16,23)(18,21)(22,24);(1,3)(2,5)(4,7)(6,8)(11,17)(15,23)(18,20)(19,24);(9,12,13,16,14,21,22,10)(11,18,19,15,17,20,24,23)Polycyclic presentation:
2^9|a^2=g,d^2=g,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=ci,d^b=di,f^b=fh,e^c=eh86 conjugacy classes — view the full character table.
| maximal subgroup of | #72339532, #72339533, #72339534, … |
| maximal normal subgroup of | #72339532, #72339533, #72339534, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 116026);