Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405389 |
| 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 | |
1253 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(5,6)(9,13)(10,22)(11,19)(14,29)(15,16)(17,20)(24,25)(27,30)(31,32);(1,2)(3,13,16,9)(4,10)(5,19)(6,8)(7,11)(12,26,15,21)(14,28,30,23)(17,24)(18,25)(20,22)(27,32,29,31);(1,3,5,15,7,16,6,12)(2,9,8,21,11,13,19,26)(4,14,17,29,18,30,20,27)(10,23,22,31,25,28,24,32);(1,4)(2,10)(3,14)(5,17)(6,20)(7,18)(8,22)(9,23)(11,25)(12,27)(13,28)(15,29)(16,30)(19,24)(21,31)(26,32)Polycyclic presentation:
2^9|b^2=h,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72319352, #72319359, #72319453 |
| maximal normal subgroup of | #72319352, #72319359, #72319453 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312585);