Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477006 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17101 |
| 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 | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12,8,13,6,31)(3,14,17,9,18,35,32,36)(4,19,23,38,24,41,22,10)(5,25,29,42,30,11,28,40)(15,45,47,60,20,33,51,43)(16,48,50,34,26,44,52,57)(21,39,55,62,56,53,27,37)(46,59,49,63,54,64,58,61);(1,3,7,17,8,18,6,32)(2,9,12,35,13,36,31,14)(4,20,23,51,24,15,22,47)(5,26,29,52,30,16,28,50)(10,33,19,43,38,45,41,60)(11,34,40,44,25,57,42,48)(21,46,55,49,56,54,27,58)(37,59,39,63,62,64,53,61);(1,4,8,24)(2,10,13,38)(3,15,18,20)(5,27,30,55)(6,23,7,22)(9,33,36,45)(11,39,25,53)(12,41,31,19)(14,43,35,60)(16,49,26,58)(17,51,32,47)(21,29,56,28)(34,61,57,63)(37,42,62,40)(44,64,48,59)(46,52,54,50);(1,5)(2,11)(3,16)(4,21)(6,29)(7,28)(8,30)(9,34)(10,37)(12,42)(13,25)(14,44)(15,46)(17,52)(18,26)(19,53)(20,54)(22,55)(23,27)(24,56)(31,40)(32,50)(33,59)(35,48)(36,57)(38,62)(39,41)(43,61)(45,64)(47,58)(49,51)(60,63)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,e^d=ei,f^d=fi,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384203);