Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #421064 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15283 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1905 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)(4,17)(5,9)(6,11)(7,15)(8,19)(10,21)(13,22)(14,23)(16,24)(18,27)(20,26)(25,31)(28,29)(30,32);(1,3,6,16)(2,7,11,23)(4,13)(5,14,21,15)(8,25)(9,12,10,24)(17,32)(18,29)(19,30)(20,31)(22,27)(26,28);(1,4)(2,8)(3,13)(5,18)(6,20)(7,22)(9,17)(10,26)(11,27)(12,28)(14,29)(15,30)(16,31)(19,21)(23,25)(24,32);(2,9)(3,14)(5,21)(8,17)(10,11)(12,24)(13,29)(15,16)(18,19)(26,27)(28,32)(30,31)Polycyclic presentation:
2^9|b^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,f^c=fi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72215972, #72216154, #72216192, … |
| maximal normal subgroup of | #72215972, #72216154, #72216192, … |
| maximal quotient of | #72237172, C2×#421064, #72252128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 328260);