Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468070 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16600 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1056 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)(4,20)(5,25)(6,14)(7,12)(8,13)(9,18)(10,23)(11,28)(16,44)(17,32)(19,41)(21,38)(22,36)(24,37)(26,35)(27,40)(29,39)(30,33)(31,46)(34,51)(42,47)(43,49)(45,56)(48,62)(50,59)(52,58)(53,55)(54,61)(57,64)(60,63);(1,3)(2,9,13,33)(4,16)(5,17)(6,18)(7,30)(8,19)(10,34,20,31)(11,32,40,59)(12,15,14,41)(21,45)(22,46)(23,47)(24,51)(25,43,39,58)(26,48)(27,49)(28,52)(29,50)(35,60,53,56)(36,42,37,44)(38,62,61,57)(54,63)(55,64);(1,4)(2,10,13,20)(3,16)(5,26)(6,24)(7,22)(8,23)(9,31,33,34)(11,38,40,61)(12,37,14,36)(15,42,41,44)(17,48)(18,51)(19,47)(21,27)(25,53,39,35)(28,54)(29,55)(30,46)(32,57,59,62)(43,60,58,56)(45,49)(50,64)(52,63);(1,5,8,29)(2,11,13,40)(3,17,19,50)(4,21,23,54)(6,28,7,27)(9,32,33,59)(10,35,20,53)(12,25,14,39)(15,43,41,58)(16,45,47,63)(18,52,30,49)(22,55,24,26)(31,56,34,60)(36,61,37,38)(42,57,44,62)(46,64,51,48)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375267);