Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #427 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #20818 |
| prime radical | C2×#41 | C2×D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #405 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
471 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2)(3,9,17,33)(4,20,22,35)(5,11)(6,14)(7,28)(8,29)(10,23,36,53)(12,26)(13,25)(15,39)(16,31,45,32)(18,47,46,59)(19,48,44,34)(21,49,52,37)(24,54,51,50)(27,38)(30,55)(40,60)(41,56)(42,58)(43,64)(57,63)(61,62);(1,3)(2,9)(4,15)(5,16)(6,17)(7,18)(8,19)(10,30)(11,31)(12,32)(13,34)(14,33)(20,39)(21,40)(22,41)(23,42)(24,43)(25,44)(26,45)(27,46)(28,47)(29,48)(35,56)(36,58)(37,57)(38,59)(49,60)(50,61)(51,62)(52,63)(53,55)(54,64);(1,4,7,23,8,24,26,52,6,22,27,53,25,51,5,21)(2,10,12,20,13,37,28,50,14,36,11,35,29,49,38,54)(3,15,18,42,19,43,45,63,17,41,46,55,44,62,16,40)(9,30,32,39,34,57,47,61,33,58,31,56,48,60,59,64);(1,5,25,27,6,26,8,7)(2,11,13,38,14,12,29,28)(3,16,44,46,17,45,19,18)(4,21,51,53,22,52,24,23)(9,31,34,59,33,32,48,47)(10,35,37,54,36,20,49,50)(15,40,62,55,41,63,43,42)(30,56,57,64,58,39,60,61)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=eh,f^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,g^b=gi,g^c=gi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72320359, #72320367, #72320377, … |
| maximal normal subgroup of | #72320359, #72320367, #72320377, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401407);