Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #499885 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #23900 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #60391 | 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 | |
1808 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)(5,13)(6,26)(7,31)(8,14)(9,18)(11,25)(12,30)(16,49)(17,34)(19,44)(20,56)(21,40)(22,59)(23,46)(24,41)(27,36)(28,39)(29,43)(32,48)(33,54)(35,55)(37,57)(38,58)(42,61)(45,53)(47,51)(50,63)(52,62)(60,64);(1,3,8,19)(2,9,14,36)(4,20,24,52)(5,17,7,32)(6,27,30,18)(10,37,41,64)(11,34,43,48)(12,44,26,15)(13,45,31,35)(16,21,51,23)(22,60,28,57)(25,55,29,53)(33,46,63,40)(38,47,42,49)(39,62,59,56)(50,61,54,58);(1,4,8,24)(2,10,14,41)(3,16,19,51)(5,23,7,21)(6,28,30,22)(9,33,36,63)(11,42,43,38)(12,39,26,59)(13,46,31,40)(15,47,44,49)(17,52,32,20)(18,54,27,50)(25,58,29,61)(34,56,48,62)(35,64,45,37)(53,57,55,60);(1,5,8,7)(2,11,14,43)(3,17,19,32)(4,21,24,23)(6,29,30,25)(9,34,36,48)(10,38,41,42)(12,31,26,13)(15,45,44,35)(16,20,51,52)(18,55,27,53)(22,61,28,58)(33,62,63,56)(37,47,64,49)(39,46,59,40)(50,60,54,57)Polycyclic presentation:
2^9|b^2=i,c^2=i,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,f^b=fh,g^b=gi,d^c=di,e^c=eh,g^c=gi,e^d=ei,f^d=fhi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72341696, #72341886, #72341975, … |
| maximal normal subgroup of | #72341696, #72341886, #72342030, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407086);