Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #448893 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15924 |
| 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 | |
875 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,14,8,11,7,3)(4,12,17,9,19,27,18,15)(5,10,21,13,22,25,23,20)(16,29,30,24,31,26,32,28);(1,3,6,11,8,14,7,2)(4,15,17,27,19,9,18,12)(5,20,21,25,22,13,23,10)(16,24,30,29,31,28,32,26);(1,4)(2,9,11,15)(3,12,14,27)(5,16)(6,17)(7,18)(8,19)(10,24,25,28)(13,26,20,29)(21,30)(22,31)(23,32);(1,5,7,21,8,22,6,23)(2,10,14,20,11,25,3,13)(4,16,18,30,19,31,17,32)(9,24,27,29,15,28,12,26)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,d^2=eg,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72318824, #72318963, #72318981, … |
| maximal normal subgroup of | #72318824, #72318963, #72318981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356089);