Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61071 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2640 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6,7,8)(9,10,13,18)(11,17,16,15)(12,22,23,21)(14,20,24,19);(1,2)(3,4)(9,11,13,16)(10,15,18,17)(12,20,23,19)(14,21,24,22);(9,12,14,17,13,23,24,15)(10,16,19,22,18,11,20,21);(2,4)(9,11)(10,17)(12,20)(13,16)(14,21)(15,18)(19,23)(22,24);(11,16)(12,23)(15,17)(21,22)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,g^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,e^c=ei,e^d=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72285046, #72285158 |
| maximal normal subgroup of | #72285046, #72285158 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6287289);