Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213852 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #2403 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,11,6)(4,12,13,7)(5,8)(10,15)(14,16)(17,18)(19,25,27,22)(20,23)(21,24)(26,31,28,30)(29,32);(1,3,5,11)(2,6,8,9)(4,10,13,16)(7,15,12,14);(1,4,5,13)(2,7,8,12)(3,10,11,16)(6,14,9,15)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32);(1,4)(2,7)(3,10)(5,13)(6,15)(8,12)(9,14)(11,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,d^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,d^c=di,g^c=gi,g^d=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285066, #72285088, #72285236, … |
| maximal normal subgroup of | #72285066, #72285088, #72285236, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121071);