Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501196 |
| 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 |
| prime radical | #23884 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1051 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)(5,25)(6,27,29,12)(7,13)(8,14)(9,18,34,52)(11,26,42,30)(16,45)(17,33)(19,46)(20,53)(21,44)(22,56,28,38)(23,39)(24,40)(31,60)(32,47,61,51)(35,50,43,59)(36,54,63,58)(37,55,41,57)(48,64)(49,62);(1,3)(2,9)(4,20)(5,17)(6,18)(7,31)(8,19)(10,36)(11,33)(12,15)(13,43)(14,34)(16,23)(21,48)(22,54)(24,49)(25,35)(26,50)(27,46)(28,58)(29,52)(30,59)(32,39)(37,64)(38,53)(40,63)(41,45)(42,60)(44,61)(47,57)(51,55)(56,62);(1,4,8,24)(2,10,14,40)(3,16,19,48)(5,23,7,21)(6,28,29,22)(9,32,34,61)(11,41,42,37)(12,38,27,56)(13,44,25,39)(15,45,46,64)(17,49,31,20)(18,51,52,47)(26,55,30,57)(33,62,60,53)(35,63,43,36)(50,54,59,58);(1,5,8,7)(2,11,14,42)(3,17,19,31)(4,21,24,23)(6,26,29,30)(9,33,34,60)(10,37,40,41)(12,25,27,13)(15,35,46,43)(16,20,48,49)(18,50,52,59)(22,55,28,57)(32,53,61,62)(36,64,63,45)(38,44,56,39)(47,54,51,58)Polycyclic presentation:
2^9|a^2=f,c^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,d^c=di,e^c=ehi,g^c=gi,f^d=fh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72303726, #72303746, #72303747, … |
| maximal normal subgroup of | #72303726, #72303746, #72303747, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408395);