Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #20338 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C42 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | C22×C42 |
| commutator | C4 | C23×C42 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C2 | C23×#97 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
2121 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,5,10,6,12,22,34,7,11,23,33,24,35,49,57)(3,13,16,38,17,27,43,32,15,37,21,48,42,8,25,28)(4,9,19,30,20,31,47,56,18,29,44,53,45,54,62,64)(14,36,40,59,41,51,61,55,39,58,46,63,60,26,50,52);(1,3)(2,8,11,27)(4,14)(5,21)(6,17)(7,15)(9,26,29,51)(10,32,33,28)(12,13,35,37)(16,23)(18,39)(19,46)(20,41)(22,25)(24,42)(30,55,53,52)(31,36,54,58)(34,48,57,38)(40,44)(43,49)(45,60)(47,50)(56,63,64,59)(61,62);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,26)(10,30)(11,29)(12,31)(13,36)(15,39)(16,40)(17,41)(21,46)(22,47)(23,44)(24,45)(25,50)(27,51)(28,52)(32,55)(33,53)(34,56)(35,54)(37,58)(38,59)(42,60)(43,61)(48,63)(49,62)(57,64);(4,18)(9,29)(14,39)(19,44)(20,45)(26,51)(30,53)(31,54)(36,58)(40,46)(41,60)(47,62)(50,61)(52,55)(56,64)(59,63);(3,15)(8,27)(13,37)(14,39)(16,21)(17,42)(25,43)(26,51)(28,32)(36,58)(38,48)(40,46)(41,60)(50,61)(52,55)(59,63)Polycyclic presentation:
2^9|a^2=f,b^2=g,f^2=h,h^2=i,b^a=bh,g^a=gi,e^b=ei,f^b=fi,d^c=di164 conjugacy classes — view the full character table.
| maximal subgroup of | #72318869, #72318884, #72319553, … |
| maximal normal subgroup of | #72318869, #72318884, #72319553, … |
| maximal quotient of | D4×#2110, #72320807, #72320969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529619);