Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7623152 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#3109 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | #33531 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1465 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)(3,9)(4,20)(5,25)(6,27)(7,32)(8,14)(10,23)(11,18)(12,16)(13,30)(15,39)(17,36)(19,52)(21,57)(22,56)(24,60)(26,37)(28,42)(29,45)(31,35)(33,43)(34,47)(38,50)(40,59)(41,62)(44,49)(46,54)(48,61)(51,64)(53,63)(55,58);(1,3)(2,9)(4,15)(5,16)(6,28)(7,19)(8,17)(10,34)(11,35)(12,25)(13,38)(14,36)(18,31)(20,54)(21,23)(22,46)(24,48)(26,49)(27,42)(29,58)(30,50)(32,52)(33,53)(37,44)(39,56)(40,57)(41,55)(43,61)(45,64)(47,59)(51,62)(60,63);(1,4)(2,10)(3,15)(5,21)(6,29)(7,33)(8,22)(9,34)(11,39)(12,43)(13,45)(14,40)(16,23)(17,46)(18,51)(19,53)(20,27)(24,30)(25,61)(26,47)(28,58)(31,62)(32,41)(35,56)(36,57)(37,63)(38,64)(42,54)(44,60)(48,50)(49,59)(52,55);(1,5,7,18,8,26,30,28)(2,11,13,25,14,42,32,37)(3,16,19,31,17,49,50,6)(4,21,33,51,22,47,24,58)(9,35,38,12,36,27,52,44)(10,39,45,61,40,54,41,63)(15,23,53,62,46,59,48,29)(20,55,60,34,56,64,43,57);(3,17)(5,26)(9,36)(11,42)(15,46)(18,28)(19,50)(21,47)(25,37)(34,57)(38,52)(39,54)(48,53)(51,58)(55,64)(61,63)Polycyclic presentation:
2^9|d^2=fh,g^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gh,h^a=hi,e^b=ei,g^b=gi,f^c=fi,g^c=gh,h^c=hi,e^d=ei,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530348);