Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #397242 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14747 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1414 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,14,8,12,7,13)(3,9,19,32,17,34,18,33)(4,20,23,10,24,38,25,37)(5,26,27,11,28,40,29,39)(15,41,44,35,21,49,43,30)(16,45,48,57,46,58,47,31)(22,52,53,50,54,36,55,60)(42,59,63,64,51,56,62,61);(1,3,7,19,8,17,6,18)(2,9,14,33,12,34,13,32)(4,21,25,43,24,15,23,44)(5,16,29,48,28,46,27,47)(10,35,38,41,37,30,20,49)(11,31,40,58,39,57,26,45)(22,51,55,62,54,42,53,63)(36,59,60,61,52,56,50,64);(1,4)(2,10,12,37)(3,15,17,21)(5,22)(6,23)(7,25)(8,24)(9,30)(11,36,39,52)(13,20,14,38)(16,42,46,51)(18,43,19,44)(26,50,40,60)(27,53)(28,54)(29,55)(31,56)(32,41)(33,49)(34,35)(45,61)(47,62,48,63)(57,59)(58,64);(1,5)(2,11)(3,16)(4,22)(6,27)(7,29)(8,28)(9,31)(10,36)(12,39)(13,26)(14,40)(15,42)(17,46)(18,47)(19,48)(20,50)(21,51)(23,53)(24,54)(25,55)(30,56)(32,45)(33,58)(34,57)(35,59)(37,52)(38,60)(41,61)(43,62)(44,63)(49,64)Polycyclic presentation:
2^9|a^2=ef,b^2=fi,c^2=e,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, 304439);