Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396123 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14730 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23439 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1406 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,11,6,9)(3,13,14,26,17,8,16,23)(4,18,19,24,20,25,21,10)(15,29,30,27,31,32,28,22);(1,3)(2,8)(4,15)(5,16)(6,14)(7,17)(9,23)(10,22)(11,13)(12,26)(18,32)(19,28)(20,31)(21,30)(24,27)(25,29);(2,9)(3,14)(5,6)(10,18)(11,12)(15,28)(16,17)(19,21)(23,26)(24,25)(29,32)(30,31);(1,4)(2,10,11,24)(3,15)(5,21)(6,19)(7,20)(8,22,13,27)(9,18,12,25)(14,28)(16,30)(17,31)(23,32,26,29)Polycyclic presentation:
2^9|a^2=f,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216071, #72216130, #72216192, … |
| maximal normal subgroup of | #72216071, #72216130, #72216192, … |
| maximal quotient of | #72237175, C2×#396123, #72252193, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303319);