Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519056 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #58997 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2454 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,9,16)(3,17,21,47)(4,22,27,48)(5,28,32,38)(6,33,34,45)(7,15,8,14)(10,30,39,20)(11,35,43,26)(12,36,44,31)(13,29,46,19)(18,37,52,62)(23,49,51,40)(24,41,57,64)(25,58,59,42)(50,54,60,63)(53,56,61,55);(1,3,8,30,9,21,7,20)(2,10,15,47,16,39,14,17)(4,23,35,52,27,51,26,18)(5,29,36,34,32,19,31,6)(11,40,48,62,43,49,22,37)(12,13,38,45,44,46,28,33)(24,25,55,60,57,59,56,50)(41,54,61,42,64,63,53,58);(1,4)(2,11)(3,18)(5,24)(6,25)(7,26)(8,35)(9,27)(10,37)(12,41)(13,42)(14,22)(15,48)(16,43)(17,49)(19,50)(20,51)(21,52)(23,30)(28,53)(29,60)(31,56)(32,57)(33,54)(34,59)(36,55)(38,61)(39,62)(40,47)(44,64)(45,63)(46,58);(1,5)(2,12)(3,19)(4,24)(6,30)(7,31)(8,36)(9,32)(10,13)(11,41)(14,28)(15,38)(16,44)(17,33)(18,50)(20,34)(21,29)(22,53)(23,25)(26,56)(27,57)(35,55)(37,42)(39,46)(40,63)(43,64)(45,47)(48,61)(49,54)(51,59)(52,60)(58,62);(1,6,8,29,9,34,7,19)(2,13,15,45,16,46,14,33)(3,5,30,36,21,32,20,31)(4,25,35,60,27,59,26,50)(10,38,47,44,39,28,17,12)(11,42,48,63,43,58,22,54)(18,24,23,55,52,57,51,56)(37,61,40,64,62,53,49,41)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=fi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,g^c=gi,h^c=hi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7426256);