Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458521 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16319 |
| 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 | |
955 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)(4,12,16,9,17,24,20,23)(5,10,18,27,19,26,21,13)(15,29,30,25,32,28,31,22);(1,3,6,11,8,14,7,2)(4,9,16,12,17,23,20,24)(5,13,18,26,19,27,21,10)(15,22,30,28,32,25,31,29);(1,4,6,16,8,17,7,20)(2,9,3,12,11,23,14,24)(5,15,18,30,19,32,21,31)(10,25,13,29,26,22,27,28);(1,5)(2,10)(3,13)(4,15)(6,18)(7,21)(8,19)(9,22)(11,26)(12,28)(14,27)(16,30)(17,32)(20,31)(23,25)(24,29)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,g^c=gi,h^c=hi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72318969, #72318981, #72318991, … |
| maximal normal subgroup of | #72318969, #72318981, #72318991, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365717);