Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511825 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #32898 |
| commutator, Frattini, cosocle | C2×#41 | C24 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2284 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)(3,14)(4,9)(5,11)(6,10)(7,13)(8,16)(12,25)(15,43)(17,41)(18,40)(19,39)(20,32)(21,31)(22,34)(23,37)(24,35)(26,44)(27,46)(28,57)(29,47)(30,49)(33,53)(36,51)(38,54)(42,48)(45,61)(50,60)(52,58)(55,62)(56,63)(59,64);(1,3,7,17)(2,8,13,29)(4,19,22,15)(5,16,23,47)(6,18,25,48)(9,30,34,26)(10,27,12,28)(11,14,37,41)(20,49,51,44)(21,50,53,45)(24,46,54,57)(31,56,33,55)(32,39,36,43)(35,40,38,42)(52,63,64,62)(58,60,59,61);(1,4,7,22)(2,9,13,34)(3,15,17,19)(5,20,23,51)(6,21,25,53)(8,26,29,30)(10,31,12,33)(11,36,37,32)(14,39,41,43)(16,44,47,49)(18,45,48,50)(24,52,54,64)(27,55,28,56)(35,59,38,58)(40,60,42,61)(46,62,57,63);(2,10)(6,25)(8,27)(9,31)(11,35)(12,13)(14,40)(18,48)(21,53)(24,54)(26,55)(28,29)(30,56)(32,58)(33,34)(36,59)(37,38)(39,60)(41,42)(43,61)(45,50)(46,57)(52,64)(62,63)Polycyclic presentation:
2^9|b^2=i,c^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,h^a=hi,c^b=ci,f^b=fhi,e^c=eh,g^d=gi,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72260417, C2×#511829, #72303715, … |
| maximal normal subgroup of | #72260417, C2×#511829, #72303715, … |
| maximal quotient of | #72260507, #72261278, #72261310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419025);