Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#546 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1246 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)(3,7,8,5)(4,6)(10,14)(12,21)(15,22)(16,17)(18,23)(19,24);(9,10,13,17)(11,18,20,15)(12,16,21,14)(19,22,24,23);(1,3)(2,5)(4,8)(6,7)(9,11,13,20)(10,15,17,18)(12,19,21,24)(14,22,16,23);(10,16)(12,21)(14,17)(15,23)(18,22)(19,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,c^b=ch,d^b=de,e^b=eh,g^c=gi,e^d=eh65 conjugacy classes — view the full character table.
| maximal subgroup of | #72339479, #72339481, #72339488, … |
| maximal normal subgroup of | #72339479, #72339481, #72339488, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 127462);