Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198789 |
| 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 | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3205 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1554 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,3,7,5,8,4,6)(9,10)(11,13)(12,20,22,16)(14,19)(15,17)(18,21,24,23);(2,6)(3,4)(7,8)(9,11,13,10,15,19,14,17)(12,18,21,16,22,24,23,20);(2,7)(3,4)(6,8)(9,12)(10,16)(11,18)(13,21)(14,23)(15,22)(17,20)(19,24);(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=eghi,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fi,g^c=ghi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72316638, #72316702, #72316704, … |
| maximal normal subgroup of | #72316638, #72316702, #72316704, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 106141);