Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #190047 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1846 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)(3,7,8,5)(4,6)(9,10)(11,19,21,15)(12,17)(13,16)(14,18)(20,24,22,23);(9,11)(10,15)(12,22)(13,20)(14,21)(16,23)(17,24)(18,19);(1,3)(2,5)(4,8)(6,7)(10,16,18,17)(15,23,19,24);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,23,19,24)Polycyclic presentation:
2^9|a^2=g,c^2=e,d^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,g^b=gh,g^c=ghi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72255780, #72256015, #72256044 |
| maximal normal subgroup of | #72255780, #72256015, #72256044 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97244);