Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #457100 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16290 |
| 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 | |
1078 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,12,7,13,6,28)(3,14,19,34,18,33,17,9)(4,10,29,37,23,38,22,41)(5,24,30,55,27,11,26,40)(15,35,20,31,45,58,44,59)(16,46,50,32,49,42,48,53)(21,36,56,39,52,62,25,54)(43,61,51,57,63,60,47,64);(1,3,7,18)(2,9)(4,20,23,44)(5,16,27,49)(6,17,8,19)(10,35)(11,32)(12,14)(13,34)(15,29,45,22)(21,51,52,47)(24,53)(25,43,56,63)(26,48,30,50)(28,33)(31,37)(36,61)(38,58)(39,57)(40,42)(41,59)(46,55)(54,64)(60,62);(1,4)(2,10)(3,15)(5,25)(6,29)(7,23)(8,22)(9,31)(11,39)(12,41)(13,38)(14,35)(16,47)(17,20)(18,45)(19,44)(21,26)(24,54)(27,56)(28,37)(30,52)(32,60)(33,58)(34,59)(36,40)(42,57)(43,48)(46,64)(49,51)(50,63)(53,61)(55,62);(1,5,6,26,7,27,8,30)(2,11,12,40,13,24,28,55)(3,16,17,48,18,49,19,50)(4,21,22,25,23,52,29,56)(9,32,14,42,34,53,33,46)(10,36,37,39,38,62,41,54)(15,43,44,47,45,63,20,51)(31,57,58,60,59,64,35,61)Polycyclic presentation:
2^9|a^2=ef,b^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi,h^c=hi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364296);