Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #476607 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17093 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
981 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,6,27,7,12,18,11)(3,13,16,8,17,41,42,31)(4,9,32,34,21,35,20,51)(5,22,25,38,26,10,28,39)(14,40,60,58,45,64,44,29)(15,36,23,55,47,53,48,30)(19,33,52,37,50,61,24,54)(43,59,49,62,56,63,46,57);(1,3,6,16,7,17,18,42)(2,8,11,13,12,31,27,41)(4,14,20,44,21,45,32,60)(5,23,25,47,26,48,28,15)(9,29,34,40,35,58,51,64)(10,36,38,30,22,53,39,55)(19,49,24,43,50,46,52,56)(33,59,37,62,61,63,54,57);(1,4,18,32,7,21,6,20)(2,9,27,51,12,35,11,34)(3,14,42,60,17,45,16,44)(5,24,28,19,26,52,25,50)(8,29,41,64,31,58,13,40)(10,37,39,33,22,54,38,61)(15,46,48,43,47,49,23,56)(30,59,36,57,55,63,53,62);(1,5)(2,10)(3,15)(4,19)(6,28)(7,26)(8,30)(9,33)(11,39)(12,22)(13,36)(14,43)(16,48)(17,47)(18,25)(20,52)(21,50)(23,42)(24,32)(27,38)(29,57)(31,55)(34,54)(35,61)(37,51)(40,63)(41,53)(44,49)(45,56)(46,60)(58,62)(59,64)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,g^d=gi47 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 383804);