Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7622907 |
| 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×Q8 | C25 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×D4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
3614 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)(4,10)(5,22)(6,9)(7,13)(8,16)(11,25)(12,27)(15,41)(17,40)(18,38)(19,35)(20,50)(21,33)(23,32)(24,34)(26,42)(28,39)(29,46)(30,44)(31,47)(36,48)(37,49)(43,59)(45,61)(51,56)(52,58)(53,62)(54,60)(55,63)(57,64);(1,3,7,18)(2,8,13,32)(4,15,19,43)(5,23,25,16)(6,26,27,17)(9,29,12,31)(10,30,35,56)(11,38,22,14)(20,51,48,44)(21,53,49,45)(24,46,28,47)(33,55,37,57)(34,40,39,42)(36,59,50,41)(52,63,54,64)(58,61,60,62);(2,9)(6,27)(8,29)(10,33)(11,39)(12,13)(14,40)(17,26)(21,49)(22,34)(24,28)(30,55)(31,32)(35,37)(36,60)(38,42)(41,61)(45,53)(46,47)(50,58)(52,54)(56,57)(59,62)(63,64);(1,4)(2,10)(3,15)(5,20)(6,21)(7,19)(8,30)(9,33)(11,36)(12,37)(13,35)(14,41)(16,44)(17,45)(18,43)(22,50)(23,51)(24,52)(25,48)(26,53)(27,49)(28,54)(29,55)(31,57)(32,56)(34,58)(38,59)(39,60)(40,61)(42,62)(46,63)(47,64);(4,19)(10,35)(15,43)(20,48)(21,49)(30,56)(33,37)(36,50)(41,59)(44,51)(45,53)(52,54)(55,57)(58,60)(61,62)(63,64)Polycyclic presentation:
2^9|b^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,f^b=fi,g^b=ghi,f^c=fh,g^c=gi,e^d=ei,g^f=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530104);