Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #22030 |
| prime radical | #3018 | C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,13,8,3,14,18,6,12,28,38,29,17,9,31)(4,19,21,39,22,43,32,47,23,10,30,35,52,15,40,44)(5,24,26,53,27,45,41,56,25,11,36,37,54,16,33,46)(20,48,50,59,51,62,57,63,49,34,55,58,64,42,60,61);(1,3,7,18,8,12,14,38,6,17,28,31,29,2,9,13)(4,19,21,39,22,43,32,47,23,10,30,35,52,15,40,44)(5,16,26,46,27,24,41,53,25,45,36,56,54,11,33,37)(20,48,50,59,51,62,57,63,49,34,55,58,64,42,60,61);(1,4)(2,10)(3,15)(5,20)(6,23)(7,30)(8,22)(9,32)(11,34)(12,19)(13,39)(14,40)(16,42)(17,43)(18,47)(21,28)(24,48)(25,49)(26,55)(27,51)(29,52)(31,44)(33,57)(35,38)(36,50)(37,59)(41,60)(45,62)(46,63)(53,58)(54,64)(56,61);(1,5)(2,11)(3,16)(4,20)(6,25)(7,26)(8,27)(9,33)(10,34)(12,24)(13,37)(14,41)(15,42)(17,45)(18,46)(19,48)(21,50)(22,51)(23,49)(28,36)(29,54)(30,55)(31,56)(32,57)(35,58)(38,53)(39,59)(40,60)(43,62)(44,61)(47,63)(52,64)Polycyclic presentation:
2^9|a^2=g,b^2=g,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ceh,f^b=fi,e^c=ei,g^c=gi,f^d=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72224962, #72320755, #72343182, … |
| maximal normal subgroup of | #72224962, #72320755, #72343182, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404003);