Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33856 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | D4×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C2 | C4⋊C4×D8 |
| prime radical | C2×#508 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #1772 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
585 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,7,14,8,15,34,13)(3,16,20,46,22,50,21,49)(4,23,6,10,25,12,35,42)(5,26,29,45,31,44,30,11)(9,36,39,17,40,32,38,19)(18,52,53,48,54,43,56,59)(24,41,33,47,28,62,58,57)(27,60,55,63,61,37,51,64);(1,3)(2,9)(4,17)(5,27)(6,32)(7,20)(8,22)(10,16)(11,43)(12,46)(13,38)(14,39)(15,40)(18,28)(19,25)(21,34)(23,49)(24,54)(26,59)(29,55)(30,51)(31,61)(33,56)(35,36)(37,41)(42,50)(44,48)(45,52)(47,64)(53,58)(57,63)(60,62);(1,4,7,6,8,25,34,35)(2,10,13,23,15,42,14,12)(3,17,20,32,22,19,21,36)(5,24,29,33,31,28,30,58)(9,16,38,49,40,50,39,46)(11,41,44,57,45,62,26,47)(18,51,53,27,54,55,56,61)(37,48,63,52,60,59,64,43);(1,5)(2,11)(3,18)(4,24)(6,33)(7,29)(8,31)(9,37)(10,41)(12,47)(13,44)(14,26)(15,45)(16,48)(17,51)(19,55)(20,53)(21,56)(22,54)(23,57)(25,28)(27,32)(30,34)(35,58)(36,61)(38,63)(39,64)(40,60)(42,62)(43,46)(49,52)(50,59)Polycyclic presentation:
2^9|a^2=g,c^2=f,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,h^a=hi,d^b=de,e^b=eh,h^b=hi,e^d=eh,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72225388, #72225399, #72225401, … |
| maximal normal subgroup of | #72225388, #72225399, #72225401, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405090);