Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500268 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24058 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| prime radical | #61676 | C2 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
815 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)(5,9)(6,29)(7,31)(8,14)(10,24)(11,19)(12,17)(13,33)(16,41)(18,42)(21,53)(22,34)(23,39)(25,52)(26,59)(27,44)(28,36)(30,43)(32,45)(35,60)(37,58)(38,47)(40,46)(48,51)(49,61)(50,55)(54,64)(56,63)(57,62);(1,3,8,19)(2,9,14,36)(4,21,24,56)(5,6,28,17)(7,32,33,18)(10,37,20,50)(11,29,15,12)(13,44,31,35)(16,46,47,25)(22,57,26,54)(23,58,48,55)(27,49,60,30)(34,52,59,40)(38,64,41,62)(39,53,51,63)(42,43,45,61);(1,4,6,23,8,24,17,48)(2,10,12,39,14,20,29,51)(3,16,5,26,19,47,28,22)(7,25,30,54,33,46,49,57)(9,34,15,38,36,59,11,41)(13,40,43,64,31,52,61,62)(18,21,27,55,32,56,60,58)(35,37,45,63,44,50,42,53);(1,5,8,28)(2,11,14,15)(3,17,19,6)(4,22,24,26)(7,27,33,60)(9,12,36,29)(10,38,20,41)(13,42,31,45)(16,23,47,48)(18,49,32,30)(21,54,56,57)(25,58,46,55)(34,51,59,39)(35,43,44,61)(37,62,50,64)(40,63,52,53)Polycyclic presentation:
2^9|b^2=i,c^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,d^b=di,e^b=ei,f^b=fh,g^b=gi,d^c=dh,f^c=fi,h^c=hi,e^d=ei,g^f=gi41 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407469);