Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13391 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2033 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C2×#3067 | C2 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(5,6,10,15,12,8,11,16)(7,17,21,31,22,34,20,33)(9,25,27,14,29,32,28,24)(13,26,30,19,23,36,18,35);(1,2)(3,4)(5,7)(6,13)(8,23)(9,26)(10,20)(11,21)(12,22)(14,31)(15,18)(16,30)(17,32)(19,28)(24,33)(25,34)(27,35)(29,36);(2,4)(5,8)(6,12)(7,18)(9,24)(10,15)(11,16)(13,20)(14,29)(17,26)(19,33)(21,23)(22,30)(25,28)(27,32)(31,35)(34,36)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dhi,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250036, #72250044, #72250104, … |
| maximal normal subgroup of | #72250036, #72250044, #72250104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 54889);