Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417863 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15258 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2676 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)(3,13)(4,9)(5,10)(6,24)(7,12)(8,16)(11,26)(14,41)(15,39)(17,40)(18,48)(19,52)(20,33)(21,55)(22,37)(23,36)(25,35)(27,56)(28,46)(29,47)(30,43)(31,51)(32,54)(34,45)(38,44)(42,59)(49,58)(50,60)(53,61)(57,64)(62,63);(1,3,6,16,7,17,26,47)(2,8,11,13,12,29,24,40)(4,14,19,43,20,44,54,56)(5,21,22,45,23,46,25,15)(9,30,32,41,33,27,52,38)(10,34,35,55,36,39,37,28)(18,49,50,63,51,64,53,42)(31,59,60,57,48,62,61,58);(1,4)(2,9)(3,14)(5,18)(6,19)(7,20)(8,27)(10,31)(11,32)(12,33)(13,38)(15,42)(16,43)(17,44)(21,49)(22,50)(23,51)(24,52)(25,53)(26,54)(28,57)(29,30)(34,62)(35,60)(36,48)(37,61)(39,59)(40,41)(45,63)(46,64)(47,56)(55,58);(1,5,7,23)(2,10,12,36)(3,15,17,45)(4,18,20,51)(6,25,26,22)(8,28,29,55)(9,31,33,48)(11,37,24,35)(13,39,40,34)(14,42,44,63)(16,46,47,21)(19,53,54,50)(27,57,30,58)(32,61,52,60)(38,59,41,62)(43,64,56,49)Polycyclic presentation:
2^9|b^2=e,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,f^b=fi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72283686, #72285925, #72286090, … |
| maximal normal subgroup of | #72283686, #72285925, #72286090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325060);