Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359528 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13867 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | #62842 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4011 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)(3,11,14,7)(4,8)(5,17,18,9)(6,10)(12,21,26,19)(13,20)(15,22,28,29)(16,23)(24,27)(25,31)(30,32);(1,3)(2,7)(4,12)(5,13)(6,14)(8,21)(9,20)(10,11)(15,25)(16,26)(17,24)(18,27)(19,23)(22,31)(28,32)(29,30);(1,4)(2,8)(3,12)(5,15)(6,16)(7,19)(9,22)(10,23)(11,21)(13,25)(14,26)(17,29)(18,28)(20,30)(24,31)(27,32);(1,5)(2,9)(3,13)(4,15)(6,18)(7,20)(8,22)(10,17)(11,24)(12,25)(14,27)(16,28)(19,30)(21,31)(23,29)(26,32)Polycyclic presentation:
2^9|a^2=h,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72232886, #72234172, #72234953, … |
| maximal normal subgroup of | #72232886, #72234172, #72234953, … |
| maximal quotient of | #72232787, #72233304, #72236600, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 266724);