Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419907 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15273 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60469 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,13,8,14,7,12)(3,15,19,34,20,42,18,9)(4,21,25,38,26,37,24,10)(5,27,29,40,31,11,30,41)(16,32,22,35,45,49,48,43)(17,39,46,60,47,57,28,33)(23,52,55,36,56,50,54,58)(44,59,51,61,53,64,63,62);(1,3)(2,9)(4,22)(5,28)(6,19)(7,18)(8,20)(10,35)(11,39)(12,42)(13,15)(14,34)(16,24)(17,29)(21,49)(23,53)(25,45)(26,48)(27,57)(30,47)(31,46)(32,37)(33,40)(36,59)(38,43)(41,60)(44,56)(50,61)(51,54)(52,62)(55,63)(58,64);(1,4,8,26)(2,10,14,38)(3,16,20,45)(5,23,31,56)(6,24,7,25)(9,32,34,49)(11,36,27,58)(12,21,13,37)(15,43,42,35)(17,44,47,53)(18,22,19,48)(28,51,46,63)(29,54,30,55)(33,59,60,64)(39,62,57,61)(40,50,41,52);(1,5,6,29,8,31,7,30)(2,11,12,40,14,27,13,41)(3,17,18,28,20,47,19,46)(4,23,24,54,26,56,25,55)(9,33,15,39,34,60,42,57)(10,36,21,50,38,58,37,52)(16,44,22,51,45,53,48,63)(32,59,43,62,49,64,35,61)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72286994, #72287280, #72287302, … |
| maximal normal subgroup of | #72286994, #72287280, #72287302, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327104);