Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395169 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#472 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,4)(9,10,13,19,15,20,14,18)(11,17,12,16,22,23,21,24);(1,2,3,4)(5,6)(7,8)(9,11,13,12,15,22,14,21)(10,16,18,17,20,24,19,23);(6,8)(9,10)(11,16)(12,17)(13,18)(14,19)(15,20)(21,23)(22,24);(9,12,13,22,15,21,14,11)(10,17,18,24,20,23,19,16)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=hi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321128, #72321159, #72321262, … |
| maximal normal subgroup of | #72321128, #72321159, #72321262, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319625);