Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #250519 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1492 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)(4,7)(6,8)(11,18)(12,13)(15,19)(16,22)(20,21)(23,24);(2,5)(3,8)(6,7)(9,10)(11,19)(12,16)(13,22)(14,17)(15,21)(18,23)(20,24);(9,11)(10,15)(12,21)(13,18)(14,20)(16,24)(17,23)(19,22);(2,5)(3,8)(6,7)(9,10)(11,15)(12,22)(13,16)(14,17)(18,24)(19,21)(20,23)Polycyclic presentation:
2^9|e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,f^d=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72249803, #72249813, #72249822, … |
| maximal normal subgroup of | #72249803, #72249813, #72249822, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 157715);