Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13077 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1893 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C2×#1893 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1277 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,5,11,8,13,7,31)(3,14,19,45,18,41,16,44)(4,20,23,38,24,10,21,40)(6,26,25,33,30,56,29,12)(9,32,36,17,37,51,35,27)(15,43,46,63,48,61,49,64)(22,54,55,42,28,57,53,39)(34,59,60,50,52,58,62,47);(1,3,6,27,7,19,29,32,8,18,30,17,5,16,25,51)(2,9,33,14,11,35,56,44,13,37,12,41,31,36,26,45)(4,15,28,47,23,49,53,58,24,48,22,50,21,46,55,59)(10,34,39,43,38,60,57,63,20,52,42,61,40,62,54,64);(1,4)(2,10)(3,15)(5,21)(6,28)(7,23)(8,24)(9,34)(11,38)(12,42)(13,20)(14,43)(16,46)(17,50)(18,48)(19,49)(22,30)(25,55)(26,54)(27,47)(29,53)(31,40)(32,58)(33,39)(35,60)(36,62)(37,52)(41,61)(44,63)(45,64)(51,59)(56,57);(1,5,8,7)(2,11,13,31)(3,16,18,19)(4,21,24,23)(6,25,30,29)(9,35,37,36)(10,38,20,40)(12,26,33,56)(14,44,41,45)(15,46,48,49)(17,32,27,51)(22,53,28,55)(34,60,52,62)(39,57,42,54)(43,63,61,64)(47,59,50,58)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249757, #72249778, #72249831, … |
| maximal normal subgroup of | #72249757, #72249778, #72249831, … |
| maximal quotient of | #72249972, C2×#494671, #72323069 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401867);