Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405374 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14912 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1255 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,7)(3,11)(5,16)(8,20)(9,19)(10,17)(14,27)(18,22)(21,29)(23,28)(26,32)(30,31);(1,2)(3,8,11,22)(4,9)(5,17)(6,10)(7,16)(12,21,24,31)(13,20,25,18)(14,28)(15,23)(19,27)(26,30,32,29);(1,3,5,13,6,11,16,25)(2,8,7,18,10,22,17,20)(4,12,14,26,15,24,27,32)(9,21,19,29,23,31,28,30);(1,4)(2,9)(3,12)(5,14)(6,15)(7,19)(8,21)(10,23)(11,24)(13,26)(16,27)(17,28)(18,29)(20,30)(22,31)(25,32)Polycyclic presentation:
2^9|b^2=h,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72338517, #72338519, #72338524 |
| maximal normal subgroup of | #72338517, #72338519, #72338524 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312570);