Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622875 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×B(4,𝔽2) |
| commutator, Frattini, cosocle | C2×D4 | C25 |
| LCS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1 | |
3660 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)(3,13)(4,9)(5,10)(6,11)(7,12)(8,17)(14,36)(15,37)(16,38)(18,29)(19,30)(20,31)(21,32)(22,33)(23,34)(24,50)(25,35)(26,41)(27,44)(28,45)(39,58)(40,59)(42,60)(43,61)(46,52)(47,55)(48,56)(49,57)(51,62)(53,63)(54,64);(1,3)(2,8)(4,14)(5,15)(6,17)(7,16)(9,26)(10,27)(11,13)(12,28)(18,39)(19,41)(20,40)(21,42)(22,44)(23,43)(24,46)(25,45)(29,51)(30,36)(31,52)(32,53)(33,37)(34,54)(35,38)(47,62)(48,63)(49,64)(50,59)(55,58)(56,60)(57,61);(1,4,7,20)(2,9,12,31)(3,14,16,40)(5,21,23,18)(6,24,25,19)(8,26,28,52)(10,32,34,29)(11,30,35,50)(13,36,38,59)(15,42,43,39)(17,46,45,41)(22,47,49,48)(27,53,54,51)(33,56,57,55)(37,60,61,58)(44,62,64,63);(1,5,7,23)(2,10,12,34)(3,15,16,43)(4,18,20,21)(6,22,25,49)(8,27,28,54)(9,29,31,32)(11,33,35,57)(13,37,38,61)(14,39,40,42)(17,44,45,64)(19,47,24,48)(26,51,52,53)(30,55,50,56)(36,58,59,60)(41,62,46,63);(3,16)(8,28)(13,38)(14,40)(15,43)(17,45)(26,52)(27,54)(36,59)(37,61)(39,42)(41,46)(44,64)(51,53)(58,60)(62,63)Polycyclic presentation:
2^9|c^2=i,d^2=i,b^a=bf,c^a=cg,h^a=hi,e^b=ei,g^b=ghi,d^c=di,f^c=fh,g^f=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72256594, #72256596, #72256597, … |
| maximal normal subgroup of | #72256594, #72256596, #72256597, … |
| maximal quotient of | #72256630, #72256648, #72312239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530073);