Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #405374 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14912 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1207 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,12)(3,13,17,34)(4,18,21,9)(5,10,23,33)(6,24,26,11)(8,25,29,16)(14,37,40,35)(15,36,42,56)(19,30,47,44)(20,31,49,45)(22,50,52,32)(27,39,43,48)(28,51,54,41)(38,57,60,58)(46,62,64,55)(53,63,61,59);(1,3)(2,8)(4,14,21,40)(5,15)(6,25)(7,17)(9,27,18,43)(10,28)(11,34)(12,29)(13,24)(16,26)(19,38,47,60)(20,48,49,39)(22,51)(23,42)(30,53,44,61)(31,37,45,35)(32,56)(33,54)(36,50)(41,52)(46,63,64,59)(55,58,62,57);(1,4,6,20,7,21,26,49)(2,9,11,31,12,18,24,45)(3,14,16,39,17,40,25,48)(5,19,22,46,23,47,52,64)(8,27,13,35,29,43,34,37)(10,30,32,55,33,44,50,62)(15,38,41,59,42,60,51,63)(28,53,36,57,54,61,56,58);(1,5)(2,10)(3,15)(4,19)(6,22)(7,23)(8,28)(9,30)(11,32)(12,33)(13,36)(14,38)(16,41)(17,42)(18,44)(20,46)(21,47)(24,50)(25,51)(26,52)(27,53)(29,54)(31,55)(34,56)(35,57)(37,58)(39,59)(40,60)(43,61)(45,62)(48,63)(49,64)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312571);