Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#2186 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
830 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,4)(5,6,8,7)(9,10,13,11)(12,15,23,18)(14,20,22,17)(16,24,19,21);(5,7,8,6)(10,15)(11,18)(12,21)(14,22)(16,17)(19,20)(23,24);(2,4)(10,11)(15,18)(16,19)(17,20);(9,11,13,10)(12,18,23,15)(14,17,22,20)(16,21,19,24)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=i,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=deh,e^b=eh,h^b=hi,d^c=di,e^d=eh,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405337);