Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501114 |
| 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 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1052 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)(4,9)(5,11,25,10)(6,12)(7,13)(8,16,31,23)(15,44)(17,45)(18,39)(19,51)(20,36)(21,34,24,37)(22,35)(26,40,57,38)(27,58)(28,42)(29,43)(30,47,60,48)(32,49,41,55)(33,52,61,53)(46,59)(50,62)(54,63,56,64);(1,3,7,18)(2,8,13,31)(4,19,20,50)(5,23,25,16)(6,27,29,17)(9,33,36,61)(10,14,11,39)(12,41,43,32)(15,28,46,22)(21,53,24,52)(26,49,57,55)(30,42,60,35)(34,62,37,51)(38,58,40,45)(44,63,59,64)(47,54,48,56);(1,4)(2,9)(3,15)(5,24)(6,28)(7,20)(8,30)(10,37)(11,34)(12,42)(13,36)(14,44)(16,48)(17,50)(18,46)(19,27)(21,25)(22,29)(23,47)(26,54)(31,60)(32,61)(33,41)(35,43)(38,63)(39,59)(40,64)(45,62)(49,52)(51,58)(53,55)(56,57);(2,10)(4,20)(5,25)(8,14)(9,34)(11,13)(12,38)(15,46)(16,23)(19,50)(22,28)(26,57)(30,59)(31,39)(32,45)(33,62)(35,63)(36,37)(40,43)(41,58)(42,64)(44,60)(49,55)(51,61)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,d^c=di,e^c=eh,g^c=gi,e^d=ei,f^d=fhi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72312286, #72312340, #72312376, … |
| maximal normal subgroup of | #72312286, #72312340, #72312376, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408311);