Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417669 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2678 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15948 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,29,6,12,8,13)(3,14,30,34,17,9,18,33)(4,10,22,37,23,38,21,41)(5,24,27,11,26,42,28,40)(15,43,19,35,45,31,48,59)(16,46,49,39,25,56,47,32)(20,51,54,61,52,55,53,36)(44,63,64,58,57,60,50,62);(1,3)(2,9)(4,19)(5,25)(6,17)(7,30)(8,18)(10,35)(11,39)(12,14)(13,34)(15,21)(16,26)(20,50)(22,45)(23,48)(24,46)(27,47)(28,49)(29,33)(31,37)(32,40)(36,60)(38,59)(41,43)(42,56)(44,54)(51,62)(52,64)(53,57)(55,58)(61,63);(1,4)(2,10)(3,15)(5,20)(6,23)(7,22)(8,21)(9,31)(11,36)(12,38)(13,41)(14,43)(16,44)(17,45)(18,48)(19,30)(24,55)(25,57)(26,52)(27,54)(28,53)(29,37)(32,58)(33,59)(34,35)(39,62)(40,61)(42,51)(46,60)(47,50)(49,64)(56,63);(1,5)(2,11)(3,16)(4,20)(6,26)(7,28)(8,27)(9,32)(10,36)(12,40)(13,42)(14,39)(15,44)(17,25)(18,49)(19,50)(21,54)(22,53)(23,52)(24,29)(30,47)(31,58)(33,56)(34,46)(35,60)(37,55)(38,61)(41,51)(43,62)(45,57)(48,64)(59,63)Polycyclic presentation:
2^9|a^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356798);