Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19456 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#386 |
| prime radical | C2×#3035 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1308 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,8)(6,7)(9,10,14,18)(11,15)(12,21,13,17)(16,22)(19,23)(20,24);(1,3,4,2,6,8,5,7)(9,11)(10,15,18,24)(12,19)(13,22)(14,20)(16,21,23,17);(9,11)(10,16)(12,19)(13,22)(14,20)(15,21)(17,24)(18,23);(1,4,6,5)(2,3,7,8)(9,12,14,13)(10,17,18,21)(11,19,20,22)(15,16,24,23)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=h,e^2=hi,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 104821);