Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6371520 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C211 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C22×#386 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3586 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5915 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)(4,19,23,37)(5,11)(6,27)(7,12)(8,13)(9,17)(10,22,34,21)(15,40,43,29)(16,39)(18,31)(20,48,49,58)(24,35)(25,52)(26,36)(28,32)(30,45)(33,51,57,50)(38,42,47,46)(41,60,61,54)(44,56)(53,55)(59,62,64,63);(1,3,6,17,8,18,7,28)(2,9,12,14,13,32,27,31)(4,15,22,46,23,43,21,42)(5,16,25,45,24,44,26,53)(10,29,37,38,34,40,19,47)(11,30,36,39,35,55,52,56)(20,41,51,63,49,61,50,62)(33,54,58,59,57,60,48,64);(1,4)(2,10)(3,15)(5,20)(6,22)(7,21)(8,23)(9,29)(11,33)(12,37)(13,34)(14,38)(16,41)(17,46)(18,43)(19,27)(24,49)(25,51)(26,50)(28,42)(30,54)(31,47)(32,40)(35,57)(36,58)(39,59)(44,61)(45,63)(48,52)(53,62)(55,60)(56,64);(1,5)(2,11)(3,16)(4,20)(6,25)(7,26)(8,24)(9,30)(10,33)(12,36)(13,35)(14,39)(15,41)(17,45)(18,44)(19,48)(21,50)(22,51)(23,49)(27,52)(28,53)(29,54)(31,56)(32,55)(34,57)(37,58)(38,59)(40,60)(42,62)(43,61)(46,63)(47,64);(5,24)(11,35)(16,44)(20,49)(25,26)(30,55)(33,57)(36,52)(39,56)(41,61)(45,53)(48,58)(50,51)(54,60)(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,f^a=fi,g^b=gi,f^c=fi,g^c=gi,h^c=hi,e^d=ei113 conjugacy classes — view the full character table.
| maximal subgroup of | #72307018, #72307024, #72307052, … |
| maximal normal subgroup of | #72307018, #72307024, #72307052, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6278716);