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 | |
1285 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)(3,14,19,37)(4,21,24,10)(5,11)(6,27)(7,31)(8,13)(9,18,34,28)(12,15)(16,40)(17,41,20,50)(22,52,25,35)(23,53,29,56)(26,39)(30,55)(32,45)(33,42,57,49)(36,54,38,43)(44,60)(46,59)(47,64)(48,51)(58,63)(61,62);(1,3,15,42,5,17,6,28,8,19,26,49,7,20,30,18)(2,9,27,50,31,57,12,37,13,34,55,41,11,33,39,14)(4,16,43,62,22,44,29,45,24,46,54,63,25,47,23,48)(10,32,53,64,52,61,38,59,21,51,56,60,35,58,36,40);(1,4)(2,10)(3,16)(5,22)(6,29)(7,25)(8,24)(9,32)(11,35)(12,38)(13,21)(14,40)(15,43)(17,44)(18,48)(19,46)(20,47)(23,30)(26,54)(27,53)(28,45)(31,52)(33,58)(34,51)(36,39)(37,59)(41,60)(42,62)(49,63)(50,64)(55,56)(57,61);(1,5,8,7)(2,11,13,31)(3,17,19,20)(4,22,24,25)(6,26,30,15)(9,33,34,57)(10,35,21,52)(12,27,39,55)(14,41,37,50)(16,44,46,47)(18,42,28,49)(23,43,29,54)(32,58,51,61)(36,56,38,53)(40,60,59,64)(45,63,48,62)Polycyclic presentation:
2^9|a^2=g,b^2=egh,d^2=i,e^2=h,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 | #72249776, #72249778, #72249827, … |
| maximal normal subgroup of | #72249776, #72249778, #72249827, … |
| maximal quotient of | #72256958, C2×#494668, #72323080 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401864);