Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #201717 |
| 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 | #37436 | C2 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C26 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1905 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)(5,10)(6,9)(11,18,25,30)(13,19,27,32)(15,21,29,22)(17,23)(24,31,26,28);(1,3)(2,7,10,20)(4,11)(5,12)(6,13)(8,18,22,31)(9,19,23,32)(14,28,21,30)(15,24)(16,26)(17,27)(25,29);(1,4)(2,8)(3,11)(5,15)(6,16)(7,18)(9,14)(10,22)(12,24)(13,26)(17,29)(19,28)(20,31)(21,23)(25,27)(30,32);(2,9)(4,15)(7,19)(8,21)(10,23)(11,24)(14,22)(16,29)(18,30)(20,32)(25,26)(28,31)Polycyclic presentation:
2^9|a^2=g,b^2=eg,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72238048, #72240651, #72251255, … |
| maximal normal subgroup of | #72238048, #72240651, #72251255, … |
| maximal quotient of | #72255656, #72255668, #72255689 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 108913);