Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60388 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C22×#120 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C22×#386 |
| prime radical | #63090 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5941 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)(3,14,18,31)(4,19)(5,11)(6,12)(7,27)(8,13)(9,17,32,26)(10,21)(15,40,44,29)(16,39,45,55)(20,47)(22,34)(23,36)(24,37)(25,51)(28,35)(30,46,54,50)(33,49)(38,42,56,43)(41,60,61,53)(48,58)(52,57)(59,62,64,63);(1,3,6,26,8,18,7,17)(2,9,12,31,13,32,27,14)(4,15,21,43,22,44,28,42)(5,16,24,50,23,45,25,46)(10,29,34,56,35,40,19,38)(11,30,37,55,36,54,51,39)(20,41,49,63,48,61,52,62)(33,53,58,64,57,60,47,59);(1,4)(2,10)(3,15)(5,20)(6,21)(7,28)(8,22)(9,29)(11,33)(12,34)(13,35)(14,38)(16,41)(17,42)(18,44)(19,27)(23,48)(24,49)(25,52)(26,43)(30,53)(31,56)(32,40)(36,57)(37,58)(39,59)(45,61)(46,62)(47,51)(50,63)(54,60)(55,64);(1,5)(2,11)(3,16)(4,20)(6,24)(7,25)(8,23)(9,30)(10,33)(12,37)(13,36)(14,39)(15,41)(17,46)(18,45)(19,47)(21,49)(22,48)(26,50)(27,51)(28,52)(29,53)(31,55)(32,54)(34,58)(35,57)(38,59)(40,60)(42,62)(43,63)(44,61)(56,64);(5,23)(11,36)(16,45)(20,48)(24,25)(30,54)(33,57)(37,51)(39,55)(41,61)(46,50)(47,58)(49,52)(53,60)(59,64)(62,63)Polycyclic presentation:
2^9|a^2=h,b^2=fh,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,f^b=fi,h^b=hi,g^c=gi,e^d=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72317429 |
| maximal normal subgroup of | #72317429 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278628);