Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418871 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2697 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15538 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1000 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)(4,16,19,28)(5,10)(6,11)(7,14)(8,20,27,18)(9,22)(13,29,30,23)(15,25)(17,26,31,32)(21,24);(1,3)(2,7)(4,17)(5,21)(6,15)(8,26)(9,25)(10,12)(11,24)(13,18)(14,22)(16,23)(19,31)(20,30)(27,32)(28,29);(1,2)(3,7)(4,8)(5,10)(6,11)(9,22)(12,21)(13,23)(14,25)(15,24)(16,20)(17,32)(18,28)(19,27)(26,31)(29,30);(1,4)(2,8)(3,13,15,30)(5,18)(6,19)(7,23,24,29)(9,16)(10,28)(11,27)(12,26,25,32)(14,17,21,31)(20,22)Polycyclic presentation:
2^9|a^2=g,d^2=f,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319988, #72320009, #72320075, … |
| maximal normal subgroup of | #72319988, #72320009, #72320075, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343010);