Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501188 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24404 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1069 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)(4,10)(5,24)(6,26,28,12)(7,13)(8,14)(9,18,34,51)(11,25,41,29)(16,44)(17,33)(19,45)(20,52)(21,39)(22,55,27,38)(23,40)(30,59)(31,43)(32,47,61,50)(35,49,42,58)(36,53,62,57)(37,54,63,56)(46,60)(48,64);(1,3,8,19)(2,9,14,34)(4,20,23,46)(5,17,7,30)(6,18,28,51)(10,36,40,62)(11,33,41,59)(12,15,26,45)(13,42,24,35)(16,31,48,21)(22,53,27,57)(25,49,29,58)(32,43,61,39)(37,44,63,64)(38,52,55,60)(47,56,50,54);(1,4)(2,10)(3,16)(5,21)(6,27)(7,31)(8,23)(9,32)(11,37)(12,38)(13,43)(14,40)(15,44)(17,46)(18,50)(19,48)(20,30)(22,28)(24,39)(25,56)(26,55)(29,54)(33,60)(34,61)(35,62)(36,42)(41,63)(45,64)(47,51)(49,53)(52,59)(57,58);(1,5,8,7)(2,11,14,41)(3,17,19,30)(4,21,23,31)(6,25,28,29)(9,33,34,59)(10,37,40,63)(12,24,26,13)(15,35,45,42)(16,46,48,20)(18,49,51,58)(22,54,27,56)(32,60,61,52)(36,44,62,64)(38,39,55,43)(47,57,50,53)Polycyclic presentation:
2^9|a^2=f,b^2=i,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,f^b=fh,g^b=gi,e^c=ehi,g^c=gi,f^d=fh,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72303720, #72303728, #72303731, … |
| maximal normal subgroup of | #72303720, #72303728, #72303731, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408389);