Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417669 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2678 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15282 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1506 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,7,13,8,12)(3,14,16,25,5,22,17,9)(4,18,20,27,21,28,23,10)(15,26,30,29,19,24,31,32);(1,3,6,17,7,5,8,16)(2,9,11,22,13,25,12,14)(4,15,20,31,21,19,23,30)(10,26,18,32,27,24,28,29);(1,4)(2,10)(3,15)(5,19)(6,23)(7,21)(8,20)(9,24)(11,28)(12,18)(13,27)(14,29)(16,31)(17,30)(22,32)(25,26);(1,5)(2,9)(3,7)(4,19)(6,16)(8,17)(10,24)(11,22)(12,14)(13,25)(15,21)(18,29)(20,30)(23,31)(26,27)(28,32)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,e^d=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318583, #72318910, #72318925, … |
| maximal normal subgroup of | #72318583, #72318910, #72318925, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328127);