Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201112 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35250 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2240 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)(3,5,7,8)(4,6)(9,10)(11,13)(12,20,22,16)(14,19)(15,17)(18,21,24,23);(1,3,4,7)(2,5)(6,8)(9,11,13,10,15,19,14,17)(12,18,21,16,22,24,23,20);(9,12)(10,16)(11,18)(13,21)(14,23)(15,22)(17,20)(19,24);(2,6)(3,7)(9,13,15,14)(10,11,17,19)(12,21,22,23)(16,18,20,24)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=hi,e^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fhi,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,g^c=ghi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279301, #72279389 |
| maximal normal subgroup of | #72279301, #72279389 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108314);