Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #62636 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C23×D8 |
| commutator | C8 | C24×C4 |
| Frattini, cosocle | C2×C8 | C25 |
| socle | C2 | C23×#102 |
| prime radical | C2×#3584 | C2 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
3913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,6,13,8,11,7,28)(3,14,17,41,18,40,29,38)(4,10,21,36,19,34,22,51)(5,23,25,37,27,53,26,12)(9,24,32,45,31,54,33,16)(15,39,44,61,42,60,52,59)(20,47,49,58,46,63,50,35)(30,48,56,62,55,64,57,43);(1,3)(2,9)(4,15)(5,24)(6,17)(7,29)(8,18)(10,30)(11,31)(12,38)(13,32)(14,23)(16,26)(19,42)(20,48)(21,44)(22,52)(25,45)(27,54)(28,33)(34,55)(35,59)(36,56)(37,41)(39,47)(40,53)(43,50)(46,64)(49,62)(51,57)(58,61)(60,63);(1,4)(2,10)(3,15)(5,20)(6,21)(7,22)(8,19)(9,30)(11,34)(12,35)(13,36)(14,39)(16,43)(17,44)(18,42)(23,47)(24,48)(25,49)(26,50)(27,46)(28,51)(29,52)(31,55)(32,56)(33,57)(37,58)(38,59)(40,60)(41,61)(45,62)(53,63)(54,64);(4,19)(10,34)(15,42)(20,46)(21,22)(30,55)(35,58)(36,51)(39,60)(43,62)(44,52)(47,63)(48,64)(49,50)(56,57)(59,61);(2,11)(9,31)(10,34)(12,37)(13,28)(14,40)(23,53)(30,55)(32,33)(35,58)(36,51)(38,41)(39,60)(47,63)(56,57)(59,61)Polycyclic presentation:
2^9|a^2=g,f^2=hi,g^2=i,h^2=i,b^a=bf,e^a=ei,f^a=fh,h^a=hi,f^b=fh,h^b=hi,d^c=di116 conjugacy classes — view the full character table.
| maximal subgroup of | #72307058, #72307063, #72307128, … |
| maximal normal subgroup of | #72307058, #72307063, #72307128, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7529699);