Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115274 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1068 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,12)(10,13)(11,15)(14,16)(17,18,21,24,22,26,20,25)(19,27,29,23,30,31,28,32);(1,3,5,10)(2,6,8,13)(4,9,11,16)(7,14,15,12)(17,19,22,30)(18,23,26,32)(20,28,21,29)(24,31,25,27);(1,4)(2,7)(3,9)(5,11)(6,12)(8,15)(10,16)(13,14)(18,24)(19,28)(20,21)(25,26)(27,31)(29,30)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,f^b=fh,d^c=di,e^c=eh,f^c=fh,g^c=gh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72312251, #72312271, #72312764, … |
| maximal normal subgroup of | #72312251, #72312271, #72312764, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22476);