Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500797 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24260 |
| 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 | |
1017 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,11,18,38)(4,19,23,50)(5,24,16,9)(6,27,25,12)(7,13,30,41)(10,33,36,22)(15,42,45,64)(17,37,29,59)(20,51,43,48)(21,54,52,49)(26,40,46,32)(28,58,47,39)(31,53,56,61)(34,60,63,44)(35,55,57,62);(1,3,6,16,8,18,25,5)(2,9,12,11,14,24,27,38)(4,15,21,43,23,45,52,20)(7,29,28,26,30,17,47,46)(10,31,35,34,36,56,57,63)(13,40,39,59,41,32,58,37)(19,48,49,42,50,51,54,64)(22,44,55,61,33,60,62,53);(1,4)(2,10)(3,15)(5,20)(6,21)(7,22)(8,23)(9,31)(11,34)(12,35)(13,19)(14,36)(16,43)(17,44)(18,45)(24,56)(25,52)(26,53)(27,57)(28,55)(29,60)(30,33)(32,48)(37,42)(38,63)(39,49)(40,51)(41,50)(46,61)(47,62)(54,58)(59,64);(1,5,25,18,8,16,6,3)(2,11,27,9,14,38,12,24)(4,20,52,45,23,43,21,15)(7,26,47,29,30,46,28,17)(10,34,57,31,36,63,35,56)(13,37,58,32,41,59,39,40)(19,42,54,48,50,64,49,51)(22,53,62,60,33,61,55,44)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,h^c=hi,g^f=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72312761, #72312762, #72312799, … |
| maximal normal subgroup of | #72312761, #72312762, #72312799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407996);