Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #503234 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #25771 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×#131 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
1092 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)(4,10)(5,9)(6,26)(7,29)(8,14)(11,18)(12,28)(13,31)(16,43)(17,40)(19,48)(20,32)(21,53)(22,38)(23,39)(24,42)(25,34)(27,41)(30,44)(33,55)(35,49)(36,46)(37,54)(45,62)(47,56)(50,61)(51,57)(52,59)(58,63)(60,64);(1,3,6,5,8,18,28,25)(2,9,12,15,14,34,26,11)(4,19,21,49,23,50,54,64)(7,30,27,55,31,17,47,24)(10,35,37,48,39,60,53,61)(13,42,41,40,29,33,56,44)(16,45,20,51,46,63,52,22)(32,58,43,57,59,62,36,38);(1,4)(2,10)(3,16)(5,20)(6,21)(7,22,31,51)(8,23)(9,32)(11,36)(12,37)(13,38,29,57)(14,39)(15,43)(17,19,30,50)(18,46)(24,49,55,64)(25,52)(26,53)(27,45,47,63)(28,54)(33,35,42,60)(34,59)(40,61,44,48)(41,58,56,62);(1,5,6,18,8,25,28,3)(2,11,12,9,14,15,26,34)(4,20,21,46,23,52,54,16)(7,24,27,30,31,55,47,17)(10,36,37,32,39,43,53,59)(13,40,41,33,29,44,56,42)(19,22,49,45,50,51,64,63)(35,57,48,62,60,38,61,58)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=e,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,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72312437, #72312455, #72312511, … |
| maximal normal subgroup of | #72312437, #72312455, #72312511, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 410430);