Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2436 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14873 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2336 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)(4,10)(5,11)(6,9)(7,12)(8,15)(14,28)(16,21)(17,24)(18,26)(19,25)(20,23)(22,29)(27,31)(30,32);(1,3)(2,8,12,23)(4,14)(5,15)(6,20)(7,16)(9,13,11,21)(10,22,26,32)(17,29)(18,31)(19,30)(24,27,25,28);(2,9,12,11)(8,21,23,13)(10,24,26,25)(22,28,32,27);(1,4)(2,10)(3,14)(5,19)(6,17)(7,18)(8,22)(9,24)(11,25)(12,26)(13,27)(15,30)(16,31)(20,29)(21,28)(23,32)Polycyclic presentation:
2^9|b^2=g,c^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,h^b=hi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72306313, #72306450, #72306462, … |
| maximal normal subgroup of | #72306313, #72306450, #72306462, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 310559);