Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13215 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1961 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1961 |
| prime radical | #60643 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1201 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)(4,21,25,54)(5,11)(6,28)(7,30)(8,13)(9,18)(10,23,37,26)(12,15)(16,45,49,62)(17,40)(19,41)(20,42)(22,55,47,39)(24,58,46,36)(27,38)(29,59)(31,57,52,64)(32,50)(33,43)(34,51)(35,48,60,63)(44,53,56,61);(1,3,15,43,7,19,29,51,8,20,27,50,5,17,6,18)(2,9,28,40,11,32,38,42,13,34,59,41,30,33,12,14)(4,22,56,48,26,49,24,57,25,47,44,63,23,16,46,64)(10,35,61,39,21,52,36,62,37,60,53,55,54,31,58,45);(1,4)(2,10,13,37)(3,16,20,49)(5,23)(6,24)(7,26)(8,25)(9,31)(11,21,30,54)(12,36,38,58)(14,39)(15,44)(17,47,19,22)(18,48,51,63)(27,56)(28,53,59,61)(29,46)(32,35)(33,60)(34,52)(40,62)(41,45)(42,55)(43,57,50,64);(1,5,8,7)(2,11,13,30)(3,17,20,19)(4,23,25,26)(6,27,29,15)(9,32,34,33)(10,21,37,54)(12,28,38,59)(14,40,42,41)(16,47,49,22)(18,50,51,43)(24,56,46,44)(31,35,52,60)(36,53,58,61)(39,62,55,45)(48,64,63,57)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fh,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ch,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72249769, #72249778, #72249836, … |
| maximal normal subgroup of | #72249769, #72249778, #72249836, … |
| maximal quotient of | #72250037, C2×#495144, #72323006, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 402340);