Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #424604 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15323 |
| 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 | |
1268 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)(3,12,14,7)(4,8)(5,9)(6,10)(11,20)(13,21,16,27)(15,23,19,22)(17,25)(18,24)(26,30)(28,31,29,32);(1,3)(2,7)(4,16)(5,19)(6,14)(8,21)(9,23)(10,12)(11,22)(13,17)(15,20)(18,28)(24,32)(25,27)(26,31)(29,30);(1,4)(2,8)(3,13,14,16)(5,18)(6,17)(7,21,12,27)(9,24)(10,25)(11,26)(15,28,19,29)(20,30)(22,31,23,32);(2,9)(5,20)(7,22)(8,24)(10,11)(12,23)(15,19)(18,30)(21,31)(25,26)(27,32)(28,29)Polycyclic presentation:
2^9|a^2=f,c^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237642, #72240435, #72240442, … |
| maximal normal subgroup of | #72237642, #72240435, #72240442, … |
| maximal quotient of | #72237658, #72251322, #72256003, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331800);