Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #449421 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15943 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C2×#41 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
672 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)(3,14)(4,10)(5,11,26,40)(6,28)(7,12)(8,13)(9,17)(15,41)(16,44,45,38)(18,32)(19,30)(20,48,51,35)(21,53)(22,36)(23,37)(24,54,55,31)(25,56,27,39)(29,33)(34,43)(42,61,49,59)(46,57,62,63)(47,58)(50,60,52,64);(1,3,6,17,8,18,7,29)(2,9,12,14,13,33,28,32)(4,19,21,43,23,47,22,15)(5,24,27,16,26,55,25,45)(10,34,36,30,37,41,53,58)(11,38,56,31,40,44,39,54)(20,49,52,46,51,42,50,62)(35,57,64,59,48,63,60,61);(1,4,7,22,8,23,6,21)(2,10,12,36,13,37,28,53)(3,15,17,19,18,43,29,47)(5,20,27,52,26,51,25,50)(9,30,32,34,33,58,14,41)(11,35,39,60,40,48,56,64)(16,42,24,46,45,49,55,62)(31,57,44,59,54,63,38,61);(1,5)(2,11,13,40)(3,16,18,45)(4,20)(6,25)(7,27)(8,26)(9,31)(10,35,37,48)(12,39,28,56)(14,38)(15,42,43,49)(17,24,29,55)(19,46,47,62)(21,50)(22,52)(23,51)(30,57)(32,44)(33,54)(34,59)(36,60,53,64)(41,61)(58,63)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=g,d^2=eg,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 356617);