Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454087 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16249 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,17,38)(4,9,20,34)(5,10,23,37)(6,24,28,11)(8,25,31,16)(14,43,46,41)(15,47,50,42)(18,53,54,32)(19,39,26,33)(21,35,55,61)(22,40,27,36)(29,51,58,45)(30,52,60,49)(44,62,64,59)(48,63,56,57);(1,3)(2,8)(4,14)(5,15)(6,25)(7,17)(9,29)(10,30)(11,38)(12,31)(13,24)(16,28)(18,44)(19,51)(20,46)(21,56)(22,52)(23,50)(26,45)(27,49)(32,57)(33,41)(34,58)(35,62)(36,42)(37,60)(39,43)(40,47)(48,55)(53,63)(54,64)(59,61);(1,4)(2,9)(3,14)(5,21)(6,26)(7,20)(8,29)(10,35)(11,39)(12,34)(13,41)(15,48)(16,51)(17,46)(18,22)(19,28)(23,55)(24,33)(25,45)(27,54)(30,59)(31,58)(32,36)(37,61)(38,43)(40,53)(42,63)(44,49)(47,57)(50,56)(52,64)(60,62);(1,5)(2,10)(3,15)(4,18)(6,27)(7,23)(8,30)(9,32)(11,40)(12,37)(13,42)(14,44)(16,52)(17,50)(19,55)(20,54)(21,26)(22,28)(24,36)(25,49)(29,57)(31,60)(33,61)(34,53)(35,39)(38,47)(41,59)(43,62)(45,56)(46,64)(48,51)(58,63)Polycyclic presentation:
2^9|a^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361284);