Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #437716 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15610 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
943 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)(3,12,14,7)(4,16)(5,11)(6,10)(8,18)(9,15,26,20)(13,22,17,27)(19,30)(21,25)(23,31,29,28)(24,32);(1,3)(2,7)(4,17,18,13)(5,20)(6,14)(8,22,16,27)(9,25)(10,12)(11,26)(15,21)(19,28,32,31)(23,24,29,30);(1,4)(2,8,10,16)(3,13)(5,19)(6,18)(7,22,12,27)(9,23,26,29)(11,24,25,30)(14,17)(15,28)(20,31)(21,32);(1,3)(2,9,10,26)(4,13)(5,15)(6,14)(7,11,12,25)(8,23,16,29)(17,18)(19,28)(20,21)(22,24,27,30)(31,32)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=ef,d^2=ef,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72285150, #72286066, #72286134, … |
| maximal normal subgroup of | #72285150, #72286066, #72286134, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344912);