Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427841 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2778 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15554 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
934 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)(3,13)(4,9,19,25)(5,11)(6,10)(7,12)(8,17)(14,23)(15,27,30,31)(16,21)(18,26,20,24)(22,28,32,29);(3,14,16,17)(8,21,23,13)(15,29,30,28)(22,27,32,31);(1,3)(2,8)(4,15)(5,14)(6,17)(7,16)(9,22)(10,21)(11,13)(12,23)(18,28)(19,30)(20,29)(24,31)(25,32)(26,27);(1,4,5,18,7,19,6,20)(2,9,10,24,12,25,11,26)(3,15,14,28,16,30,17,29)(8,22,21,31,23,32,13,27)Polycyclic presentation:
2^9|a^2=fg,b^2=e,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72318876, #72318916, #72318981, … |
| maximal normal subgroup of | #72318876, #72318916, #72318981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343469);