Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15263 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23414 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1636 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)(3,15)(4,10)(5,26)(6,13)(7,12)(8,14)(9,18)(11,30)(16,35)(17,32)(19,34)(20,33)(21,48)(22,50)(23,42)(24,37)(25,38)(27,56)(28,41)(29,40)(31,44)(36,51)(39,47)(43,61)(45,59)(46,54)(49,64)(52,55)(53,62)(57,60)(58,63);(1,3,8,20)(2,9,14,34)(4,21,25,44)(5,27,29,47)(6,19,7,18)(10,35,38,59)(11,39,41,56)(12,33,13,15)(16,23,45,24)(17,30,46,28)(22,43,52,63)(26,54,40,32)(31,42,48,37)(36,61,62,58)(49,53,57,51)(50,60,55,64);(1,4)(2,10)(3,16)(5,22)(6,23)(7,24)(8,25)(9,31)(11,36)(12,37)(13,42)(14,38)(15,35)(17,43)(18,44)(19,21)(20,45)(26,55)(27,57)(28,51)(29,52)(30,53)(32,58)(33,59)(34,48)(39,60)(40,50)(41,62)(46,63)(47,49)(54,61)(56,64);(1,5,6,28,8,29,7,30)(2,11,12,40,14,41,13,26)(3,17,18,27,20,46,19,47)(4,22,23,51,25,52,24,53)(9,32,15,39,34,54,33,56)(10,36,37,50,38,62,42,55)(16,43,44,57,45,63,21,49)(31,58,35,60,48,61,59,64)Polycyclic presentation:
2^9|b^2=i,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 325668);