Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #471449 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16797 |
| 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 | |
1057 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,15,19,42)(4,10,24,38)(5,25,28,40)(6,13,7,12)(9,30,33,18)(11,27,41,29)(16,43,47,34)(17,32,50,60)(20,52,46,31)(21,39,54,62)(22,37,23,36)(26,35,55,56)(44,51,59,49)(45,57,64,63)(48,61,53,58);(1,3,7,30,8,19,6,18)(2,9,13,42,14,33,12,15)(4,20,22,47,24,46,23,16)(5,17,27,49,28,50,29,51)(10,34,36,52,38,43,37,31)(11,32,40,59,41,60,25,44)(21,53,55,64,54,48,26,45)(35,61,62,63,56,58,39,57);(1,4)(2,10)(3,16)(5,26)(6,23)(7,22)(8,24)(9,31)(11,39)(12,37)(13,36)(14,38)(15,43)(17,48)(18,46)(19,47)(20,30)(21,27)(25,56)(28,55)(29,54)(32,58)(33,52)(34,42)(35,40)(41,62)(44,63)(45,49)(50,53)(51,64)(57,59)(60,61);(1,5)(2,11)(3,17)(4,21)(6,29)(7,27)(8,28)(9,32)(10,35)(12,25)(13,40)(14,41)(15,44)(16,45)(18,51)(19,50)(20,53)(22,55)(23,26)(24,54)(30,49)(31,57)(33,60)(34,61)(36,62)(37,39)(38,56)(42,59)(43,58)(46,48)(47,64)(52,63)Polycyclic presentation:
2^9|a^2=i,b^2=egh,e^2=i,g^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,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,e^d=ei,f^d=fi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 378646);