Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #235012 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#392 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1798 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,22)(18,21)(19,23)(20,24);(1,2)(3,8)(4,6)(5,7)(12,19)(14,21)(15,23)(17,20);(3,7)(5,8)(9,11)(10,14)(12,20)(13,18)(15,24)(16,21)(17,19)(22,23);(2,6)(5,8)(9,12,13,19)(10,15,16,23)(11,17,18,20)(14,22,21,24)Polycyclic presentation:
2^9|d^2=hi,e^2=h,b^a=be,c^a=cf,e^a=eh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dhi,e^c=ei56 conjugacy classes — view the full character table.
| maximal subgroup of | #72286980, #72287122, #72287135 |
| maximal normal subgroup of | #72286980, #72287122, #72287135 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 142214);