Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232766 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#127 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#392 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1466 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)(3,6)(4,8)(5,7)(9,10)(11,17)(12,20)(13,16)(14,19)(15,22)(18,24)(21,23);(2,6)(3,4)(7,8)(14,23)(17,24);(1,2)(3,6)(4,8)(5,7)(9,11)(10,14)(12,21)(13,18)(15,24)(16,23)(17,20)(19,22);(2,7)(6,8)Polycyclic presentation:
2^9|e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,g^a=gi,c^b=cg,e^b=eh,g^b=gh,d^c=dh,e^c=ei,f^c=fhi,g^c=gh68 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#232766, #72249813, #72249818, … |
| maximal normal subgroup of | C2×#232766, #72249813, #72249818, … |
| maximal quotient of | #72211651, #72211656, #72211663, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 139962);