Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60372 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
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 | C22×#577 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
6221 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,14,18,15,11,13,17)(12,16,22,24,20,19,21,23);(1,2,3,4)(9,11,15,10)(12,19,20,16)(13,17,14,18)(21,23,22,24);(6,8)(9,10,15,11)(12,16,20,19)(13,18,14,17)(21,24,22,23);(9,12)(10,16)(11,19)(13,21)(14,22)(15,20)(17,23)(18,24);(12,20)(16,19)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=fhi,c^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^b=fi,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei122 conjugacy classes — view the full character table.
| maximal subgroup of | #72284953 |
| maximal normal subgroup of | #72284953 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278625);