Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2436 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14776 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,9)(5,10)(6,23)(7,12)(8,16)(11,14)(15,27)(17,32)(18,29)(19,26)(20,30)(21,28)(22,31)(24,25);(1,3,14,19,7,17,6,16)(2,8,23,32,12,26,11,13)(4,18,20,24,21,5,15,22)(9,25,27,29,28,31,30,10);(1,4,14,20,7,21,6,15)(2,9,23,27,12,28,11,30)(3,5,19,22,17,18,16,24)(8,25,32,29,26,31,13,10);(1,5)(2,10)(3,15)(4,19)(6,24)(7,18)(8,9)(11,31)(12,29)(13,30)(14,22)(16,21)(17,20)(23,25)(26,28)(27,32)Polycyclic presentation:
2^9|b^2=ef,c^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306453, #72306456, #72306473, … |
| maximal normal subgroup of | #72306453, #72306456, #72306473, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 306319);