Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7211754 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #45599 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #3018 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,28,7,13,6,12)(3,14,26,9,18,38,17,32)(4,10,27,39,21,35,20,34)(5,22,29,37,25,11,24,54)(15,42,48,30,45,40,44,58)(16,36,23,53,47,41,46,31)(19,49,56,60,52,33,51,62)(43,59,50,64,63,61,55,57);(1,3)(2,9)(4,15)(5,23)(6,26)(7,18)(8,17)(10,30)(11,36)(12,38)(13,32)(14,28)(16,29)(19,50)(20,48)(21,45)(22,41)(24,47)(25,46)(27,44)(31,54)(33,59)(34,40)(35,58)(37,53)(39,42)(43,56)(49,61)(51,63)(52,55)(57,62)(60,64);(1,4,6,27,7,21,8,20)(2,10,28,34,13,35,12,39)(3,15,26,44,18,45,17,48)(5,19,24,56,25,52,29,51)(9,30,14,40,32,58,38,42)(11,33,54,60,22,49,37,62)(16,43,23,55,47,63,46,50)(31,57,41,59,53,64,36,61);(1,5,6,24,7,25,8,29)(2,11,28,54,13,22,12,37)(3,16,26,23,18,47,17,46)(4,19,27,56,21,52,20,51)(9,31,14,41,32,53,38,36)(10,33,34,60,35,49,39,62)(15,43,44,55,45,63,48,50)(30,57,40,59,58,64,42,61);(1,3)(2,9)(4,15)(5,16)(6,26)(7,18)(8,17)(10,30)(11,31)(12,38)(13,32)(14,28)(19,43)(20,48)(21,45)(22,53)(23,24)(25,47)(27,44)(29,46)(33,57)(34,40)(35,58)(36,37)(39,42)(41,54)(49,64)(50,51)(52,63)(55,56)(59,60)(61,62)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,f^b=fi,g^b=gi,f^c=fi,g^c=gi,h^d=hi,f^e=fi,g^e=gi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7118951);