Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395169 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,16,19,17,21,25,20)(15,22,24,18,23,27,28,26);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(15,23)(18,26)(22,27)(24,28);(2,4)(5,8,9,7)(6,10,11,12)(13,14)(15,18)(16,20)(17,21)(19,25)(22,24)(23,26)(27,28);(13,15,16,24,17,23,25,28)(14,18,20,22,21,26,19,27)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=e,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=di,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321658, #72321668, #72321742, … |
| maximal normal subgroup of | #72321658, #72321668, #72321742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 319863);