Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #473463 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16866 |
| 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 | |
835 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,8,13,7,29)(3,14,28,24,16,9,17,20)(4,18,21,30,23,31,22,10)(5,11,27,32,25,19,26,15);(1,3)(2,9,13,14)(4,19,23,11)(5,10)(6,28)(7,17)(8,16)(12,20,29,24)(15,22,32,21)(18,27)(25,30)(26,31);(1,4)(2,10)(3,15)(5,24)(6,22)(7,21)(8,23)(9,26)(11,17)(12,31)(13,30)(14,27)(16,32)(18,29)(19,28)(20,25);(1,5,6,27,8,25,7,26)(2,11,12,32,13,19,29,15)(3,10,28,18,16,30,17,31)(4,20,21,14,23,24,22,9)Polycyclic presentation:
2^9|a^2=ef,b^2=fgh,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72339677, #72339678, #72339684 |
| maximal normal subgroup of | #72339677, #72339678, #72339684 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380659);