Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455577 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16266 |
| 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 | |
1095 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)(3,15)(4,20)(5,26)(6,14)(7,12)(8,13)(9,18)(10,23)(11,29)(16,32)(17,33)(19,41)(21,53)(22,55)(24,36)(25,37)(27,38)(28,40)(30,39)(31,34)(35,57)(42,45)(43,47)(44,60)(46,59)(48,61)(49,62)(50,51)(52,56)(54,64)(58,63);(1,3,6,18,8,19,7,31)(2,9,12,15,13,34,14,41)(4,21,23,50,25,16,24,45)(5,17,29,48,30,49,28,47)(10,32,20,51,36,53,37,42)(11,33,26,43,40,62,39,61)(22,54,56,46,57,44,27,58)(35,59,52,64,55,63,38,60);(1,4,8,25)(2,10,13,36)(3,16,19,21)(5,27,30,56)(6,24,7,23)(9,32,34,53)(11,38,40,52)(12,37,14,20)(15,42,41,51)(17,46,49,58)(18,50,31,45)(22,29,57,28)(26,55,39,35)(33,60,62,64)(43,63,61,59)(44,48,54,47);(1,5)(2,11)(3,17)(4,22)(6,29)(7,28)(8,30)(9,33)(10,35)(12,26)(13,40)(14,39)(15,43)(16,44)(18,48)(19,49)(20,52)(21,54)(23,56)(24,27)(25,57)(31,47)(32,59)(34,62)(36,55)(37,38)(41,61)(42,60)(45,58)(46,50)(51,64)(53,63)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 362774);