Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #404008 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14885 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2612 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)(3,15)(4,19)(5,22)(6,14)(7,12)(8,13)(9,16)(10,20)(11,23)(17,29)(18,28)(21,32)(24,31)(25,27)(26,30);(1,3)(2,9)(4,5)(6,17)(7,16)(8,18)(10,22)(11,19)(12,28)(13,29)(14,15)(20,24)(21,25)(23,26)(27,30)(31,32);(1,4)(2,10)(3,5)(6,26)(7,20)(8,21)(9,27)(11,28)(12,32)(13,30)(14,19)(15,31)(16,24)(17,23)(18,25)(22,29);(1,5)(2,11,13,31)(3,4)(6,24)(7,23)(8,25)(9,19,29,32)(10,28,30,15)(12,27,14,22)(16,26)(17,20)(18,21)Polycyclic presentation:
2^9|d^2=g,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,h^b=hi,e^c=ei,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72233772, #72233773, #72319342, … |
| maximal normal subgroup of | #72233772, #72233773, #72319342, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311204);