Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449490 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15947 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3270 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)(3,13)(4,18,22,28)(5,16)(6,15)(7,11)(8,12)(9,21,27,20)(10,24)(14,25,30,19)(17,23)(26,32,31,29);(1,3)(2,5)(4,14)(6,10)(7,16)(8,17)(9,25)(11,23)(12,24)(13,15)(18,32)(19,27)(20,31)(21,26)(22,30)(28,29);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,26)(11,28)(12,27)(13,29)(15,18)(16,31)(17,30)(23,32)(24,25);(1,5)(2,10)(3,15)(4,19)(6,23)(7,13)(8,24)(9,26)(11,17)(12,16)(14,18)(20,29)(21,32)(22,25)(27,31)(28,30)Polycyclic presentation:
2^9|a^2=hi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215968, #72216013, #72216112, … |
| maximal normal subgroup of | #72215968, #72216013, #72216112, … |
| maximal quotient of | C2×#449491, #72255851, #72265720, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356687);