Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3290 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | D42 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | #22842 |
| prime radical | #3018 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
724 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,13,33,45,9,15,34,46,6,17,32,47)(3,16,20,24,11,10,35,25,21,37,53,23,19,38,52,4)(5,12,30,43,28,42,49,63,31,44,59,64,29,41,58,61)(18,48,50,54,39,36,60,27,51,40,62,55,26,56,57,22);(1,3,7,20,8,11,33,35,9,21,34,53,6,19,32,52)(2,10,14,25,13,37,45,23,15,38,46,4,17,16,47,24)(5,26,30,57,28,18,49,50,31,39,59,60,29,51,58,62)(12,40,43,55,42,56,63,22,44,48,64,54,41,36,61,27);(1,4)(2,11)(3,17)(5,27)(6,23)(7,16)(8,24)(9,25)(10,33)(12,26)(13,21)(14,35)(15,19)(18,42)(20,47)(22,31)(28,55)(29,54)(30,40)(32,38)(34,37)(36,58)(39,44)(41,51)(43,57)(45,53)(46,52)(48,59)(49,56)(50,63)(60,64)(61,62);(1,5)(2,12)(3,18)(4,22)(6,29)(7,30)(8,28)(9,31)(10,36)(11,39)(13,42)(14,43)(15,44)(16,48)(17,41)(19,26)(20,50)(21,51)(23,55)(24,54)(25,27)(32,58)(33,49)(34,59)(35,60)(37,40)(38,56)(45,63)(46,64)(47,61)(52,57)(53,62)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,d^b=de,e^b=ei,d^c=di,f^c=fi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72225458 |
| maximal normal subgroup of | #72225458 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405303);