Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499876 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #35346 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
3789 conjugacy classes of subgroups; only the 576 normal subgroups are shown.
This group has 3789 conjugacy classes of subgroups — too many to draw. Only its 576 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,8)(4,9)(5,10)(6,11)(7,12)(13,24)(14,25)(15,26)(16,27)(17,29)(18,30)(19,31)(20,33)(21,34)(22,47)(23,35)(28,38)(32,42)(36,48)(37,49)(39,51)(40,52)(41,54)(43,53)(44,55)(45,62)(46,57)(50,59)(56,61)(58,63)(60,64);(1,3,7,16)(2,8,12,27)(4,13,19,38)(5,14,21,40)(6,22,23,15)(9,28,31,24)(10,25,34,52)(11,26,35,47)(17,36,42,59)(18,43,44,37)(20,45,46,39)(29,50,32,48)(30,53,55,49)(33,51,57,62)(41,60,61,58)(54,64,56,63);(1,4,7,19)(2,9,12,31)(3,13,16,38)(5,17,21,42)(6,18,23,44)(8,24,27,28)(10,32,34,29)(11,30,35,55)(14,36,40,59)(15,37,22,43)(20,41,46,61)(25,50,52,48)(26,49,47,53)(33,56,57,54)(39,58,45,60)(51,64,62,63);(1,5)(2,10)(3,14)(4,17)(6,20)(7,21)(8,25)(9,29)(11,33)(12,34)(13,36)(15,39)(16,40)(18,41)(19,42)(22,45)(23,46)(24,48)(26,51)(27,52)(28,50)(30,54)(31,32)(35,57)(37,58)(38,59)(43,60)(44,61)(47,62)(49,63)(53,64)(55,56);(1,6)(2,11)(3,15)(4,18)(5,20)(7,23)(8,26)(9,30)(10,33)(12,35)(13,37)(14,39)(16,22)(17,41)(19,44)(21,46)(24,49)(25,51)(27,47)(28,53)(29,54)(31,55)(32,56)(34,57)(36,58)(38,43)(40,45)(42,61)(48,63)(50,64)(52,62)(59,60)Polycyclic presentation:
2^9|b^2=i,c^2=i,b^a=bf,c^a=cg,h^a=hi,c^b=ch,e^b=ehi,d^c=dh,f^c=fi,g^d=gi,f^e=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72209847, #72209848, C2×#6499877, … |
| maximal normal subgroup of | #72209847, #72209848, C2×#6499877, … |
| maximal quotient of | #72210036, #72227298, #72227665, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407073);