Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14718 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,20,33)(4,21,25,38)(5,27,29,40)(6,13,7,12)(9,19,34,18)(10,26,37,24)(11,30,39,28)(16,31,22,35)(17,32,48,58)(23,51,53,36)(41,44,57,45)(42,46,55,47)(43,62,50,61)(49,54,60,52)(56,63,59,64);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,35)(11,32)(12,33)(13,15)(14,34)(16,25)(21,41)(23,50)(24,44)(26,45)(27,55)(28,47)(29,48)(30,46)(31,37)(36,59)(38,57)(39,58)(40,42)(43,53)(49,61)(51,56)(52,63)(54,64)(60,62);(1,4,6,24,8,25,7,26)(2,10,12,21,14,37,13,38)(3,16,18,44,20,22,19,45)(5,23,28,52,29,53,30,54)(9,31,15,41,34,35,33,57)(11,36,27,49,39,51,40,60)(17,43,46,63,48,50,47,64)(32,56,42,61,58,59,55,62);(1,5)(2,11)(3,17)(4,23)(6,28)(7,30)(8,29)(9,32)(10,36)(12,27)(13,40)(14,39)(15,42)(16,43)(18,46)(19,47)(20,48)(21,49)(22,50)(24,52)(25,53)(26,54)(31,56)(33,55)(34,58)(35,59)(37,51)(38,60)(41,61)(44,63)(45,64)(57,62)Polycyclic presentation:
2^9|a^2=i,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^c=gi,h^d=hi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301877);