Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417790 |
| 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 | |
2655 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)(3,15)(4,20)(5,11)(6,13)(7,12)(8,14)(9,18)(10,22)(16,30)(17,43)(19,40)(21,52)(23,41)(24,57)(25,38)(26,37)(27,39)(28,32)(29,35)(31,49)(33,47)(34,54)(36,48)(42,46)(44,59)(45,50)(51,55)(53,62)(56,61)(58,64)(60,63);(1,3,6,18,8,19,7,28)(2,9,12,15,14,32,13,40)(4,16,22,45,23,47,29,46)(5,24,25,48,27,49,26,17)(10,33,20,50,35,30,41,42)(11,36,37,57,39,43,38,31)(21,53,54,64,55,63,56,44)(34,60,52,58,61,62,51,59);(1,4)(2,10)(3,16)(5,21)(6,22)(7,29)(8,23)(9,30)(11,34)(12,20)(13,41)(14,35)(15,42)(17,44)(18,45)(19,47)(24,53)(25,54)(26,56)(27,55)(28,46)(31,58)(32,33)(36,62)(37,52)(38,51)(39,61)(40,50)(43,60)(48,64)(49,63)(57,59);(1,5,8,27)(2,11,14,39)(3,17,19,48)(4,21,23,55)(6,26,7,25)(9,31,32,57)(10,34,35,61)(12,38,13,37)(15,43,40,36)(16,44,47,64)(18,49,28,24)(20,51,41,52)(22,56,29,54)(30,58,33,59)(42,60,50,62)(45,63,46,53)Polycyclic presentation:
2^9|b^2=e,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,f^c=fi,h^c=hi,e^d=ei,f^d=fi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72306344, #72306459, #72306467, … |
| maximal normal subgroup of | #72306344, #72306459, #72306467, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324987);