Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201678 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1898 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)(3,7,12,20)(4,14,16,8)(5,9)(6,10)(11,19,25,32)(13,21,27,30)(15,23,29,22)(17,18)(24,31,26,28);(1,3,6,13)(2,7,18,30)(4,11,16,26)(5,12,17,27)(8,19,23,32)(9,20,10,21)(14,28,22,31)(15,24,29,25);(1,4)(2,8)(3,11)(5,15)(6,16)(7,19)(9,22)(10,14)(12,24)(13,26)(17,29)(18,23)(20,31)(21,28)(25,27)(30,32);(4,15)(8,22)(11,24)(14,23)(16,29)(19,31)(25,26)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=egi,b^a=be,c^a=cf,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72261138, #72261150, #72261169, … |
| maximal normal subgroup of | #72261138, #72261150, #72261169, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108877);