Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418849 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15266 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3026 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2131 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)(3,11,14,7)(4,15,17,8)(5,9)(6,10)(12,24)(13,20,27,22)(16,28,30,21)(18,23)(19,26)(25,31)(29,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,22)(10,11)(15,24)(16,29)(17,26)(18,27)(20,23)(21,31)(25,30)(28,32);(1,4)(2,8)(3,12)(5,16)(6,17)(7,19)(9,21)(10,15)(11,24)(13,25)(14,26)(18,30)(20,31)(22,32)(23,28)(27,29);(1,5)(2,9)(3,13)(4,16)(6,18)(7,20)(8,21)(10,23)(11,22)(12,25)(14,27)(15,28)(17,30)(19,31)(24,32)(26,29)Polycyclic presentation:
2^9|a^2=ef,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,h^b=hi,e^c=ei,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237732, #72237746, #72237754, … |
| maximal normal subgroup of | #72237732, #72237746, #72237754, … |
| maximal quotient of | #72240603, #72265861, #72265863, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326045);