Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442595 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#535 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2408 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,8)(4,17)(5,10)(6,23)(7,12)(9,20)(11,24)(13,35)(14,26)(15,27)(16,42)(18,44)(19,45)(21,33)(22,50)(25,38)(28,41)(29,53)(30,46)(31,48)(32,39)(34,49)(36,57)(37,43)(40,56)(47,62)(51,59)(52,60)(54,64)(55,63)(58,61);(1,3)(2,8)(4,13)(5,14)(6,16)(7,15)(9,29)(10,32)(11,28)(12,27)(17,43)(18,36)(19,37)(20,38)(21,39)(22,40)(23,42)(24,41)(25,31)(26,33)(30,51)(34,56)(35,45)(44,57)(46,59)(47,58)(48,53)(49,60)(50,52)(54,55)(61,62)(63,64);(1,4,6,20,7,19,24,48)(2,9,11,17,12,31,23,45)(3,13,16,38,15,37,41,53)(5,18,22,46,21,47,49,63)(8,25,28,35,27,29,42,43)(10,30,34,44,33,55,50,62)(14,36,40,59,39,58,60,64)(26,51,52,57,32,54,56,61);(1,5)(2,10)(3,14)(4,18)(6,22)(7,21)(8,26)(9,30)(11,34)(12,33)(13,36)(15,39)(16,40)(17,44)(19,47)(20,46)(23,50)(24,49)(25,51)(27,32)(28,52)(29,54)(31,55)(35,57)(37,58)(38,59)(41,60)(42,56)(43,61)(45,62)(48,63)(53,64);(3,15)(8,27)(13,37)(14,39)(16,41)(25,29)(26,32)(28,42)(35,43)(36,58)(38,53)(40,60)(51,54)(52,56)(57,61)(59,64)Polycyclic presentation:
2^9|c^2=g,g^2=i,b^a=bf,c^a=cg,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72249827, #72249829, #72249830, … |
| maximal normal subgroup of | #72249827, #72249829, #72249830, … |
| maximal quotient of | #72257114, #72323003 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349933);