Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62820 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2354 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,12)(3,13,17,30)(4,7,11,6)(5,19,23,18)(9,25,27,15)(10,22,29,21)(14,20,26,32)(16,28,24,31);(1,3)(2,9)(4,14)(5,20)(6,24)(7,16)(8,17)(10,28)(11,26)(12,27)(13,19)(15,21)(18,30)(22,25)(23,32)(29,31);(1,4,7,12)(2,8,11,6)(3,14,16,27)(5,19,21,10)(9,17,26,24)(13,15,28,20)(18,22,29,23)(25,31,32,30);(5,21)(10,19)(13,28)(15,20)(18,29)(22,23)(25,32)(30,31)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213821, #72237988, #72252033, … |
| maximal normal subgroup of | #72213821, #72237988, #72252033, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253663);