Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12778 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1790 |
| commutator | C2×Q8 | C2×C42 |
| Frattini, cosocle | C22×#41 | C23 |
| socle | C2 | #10745 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
952 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2)(3,15,19,42)(4,20)(5,24,26,11)(6,13)(7,12)(8,14)(9,17)(10,22)(16,43,46,62)(18,51,27,41)(21,54,56,53)(23,38)(25,37,29,39)(28,35)(30,58)(31,49)(32,48)(33,50)(34,57,36,55)(40,47,59,45)(44,52)(60,64)(61,63);(1,3,7,18,8,19,6,27)(2,9,13,32,14,33,12,31)(4,16,22,47,23,46,28,45)(5,17,25,49,26,50,29,48)(10,30,20,52,35,60,38,61)(11,15,37,41,24,42,39,51)(21,44,57,58,56,63,55,64)(34,59,53,43,36,40,54,62);(1,4)(2,10)(3,16)(5,21)(6,28)(7,22)(8,23)(9,30)(11,36)(12,38)(13,20)(14,35)(15,40)(17,44)(18,47)(19,46)(24,34)(25,57)(26,56)(27,45)(29,55)(31,61)(32,52)(33,60)(37,54)(39,53)(41,62)(42,59)(43,51)(48,64)(49,58)(50,63)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=i,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gi,h^a=hi,e^b=ehi,f^b=fhi,d^c=dh,e^c=ei,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72317333, #72317395, #72317399, … |
| maximal normal subgroup of | #72317333, #72317395, #72317399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53854);