Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #460625 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16368 |
| 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 | |
987 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,11,5,12,7,9)(3,13,16,24,17,8,14,23)(4,18,21,10,19,25,20,26)(15,29,28,22,31,32,30,27);(1,3,7,16,5,17,6,14)(2,8,11,24,12,13,9,23)(4,15,21,30,19,31,20,28)(10,22,18,32,26,27,25,29);(2,9)(3,14)(4,19)(6,7)(8,13)(10,25)(11,12)(15,28)(16,17)(18,26)(29,32)(30,31);(1,4)(2,10)(3,15)(5,19)(6,20)(7,21)(8,22)(9,25)(11,18)(12,26)(13,27)(14,28)(16,30)(17,31)(23,29)(24,32)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,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,e^b=ei,f^b=fi,d^c=de,f^c=fi,g^c=gi,h^c=hi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237066, #72237068, #72237090, … |
| maximal normal subgroup of | #72237066, #72237068, #72237090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367821);