Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455816 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16270 |
| 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 | |
1064 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)(4,21)(5,27)(6,13)(7,12)(8,14)(9,18)(10,24)(11,28)(16,43)(17,47)(19,33)(20,34)(22,31)(23,36)(25,37)(26,39)(29,38)(30,40)(32,51)(35,45)(41,46)(42,50)(44,60)(48,59)(49,57)(52,55)(53,63)(54,58)(56,62)(61,64);(1,3,6,19,8,18,7,20)(2,9,12,33,14,15,13,34)(4,22,24,16,26,46,25,45)(5,17,28,50,30,49,29,51)(10,35,21,41,37,43,39,31)(11,32,27,57,38,42,40,47)(23,53,54,44,56,64,55,48)(36,61,52,60,62,63,58,59);(1,4)(2,10)(3,16,18,45)(5,23)(6,25)(7,24)(8,26)(9,31,15,41)(11,36)(12,39)(13,21)(14,37)(17,48,49,44)(19,22,20,46)(27,58)(28,55)(29,54)(30,56)(32,60,42,59)(33,43,34,35)(38,62)(40,52)(47,63,57,61)(50,64,51,53);(1,5,8,30)(2,11,14,38)(3,17,18,49)(4,23,26,56)(6,28,7,29)(9,32,15,42)(10,36,37,62)(12,27,13,40)(16,44,45,48)(19,50,20,51)(21,52,39,58)(22,53,46,64)(24,54,25,55)(31,59,41,60)(33,57,34,47)(35,61,43,63)Polycyclic presentation:
2^9|b^2=f,c^2=e,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=de,f^c=fi,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363013);