Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6442700 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2408 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,8,13)(3,14,19,33)(4,20,23,38)(5,11,27,42)(6,28,7,12)(9,18,34,17)(10,29,37,24)(15,31,46,50)(16,44,49,39)(21,35,47,43)(22,51,53,60)(25,32,48,57)(26,41,30,40)(36,55,59,54)(45,56,62,64)(52,58,63,61);(1,3)(2,9)(4,21)(5,25)(6,18)(7,17)(8,19)(10,35)(11,39)(12,33)(13,34)(14,28)(15,24)(16,26)(20,50)(22,45)(23,47)(27,48)(29,46)(30,49)(31,38)(32,41)(36,56)(37,43)(40,57)(42,44)(51,58)(52,55)(53,62)(54,63)(59,64)(60,61);(1,4,6,24,8,23,7,29)(2,10,12,20,13,37,28,38)(3,15,18,47,19,46,17,21)(5,22,26,54,27,53,30,55)(9,31,33,35,34,50,14,43)(11,36,40,51,42,59,41,60)(16,45,48,63,49,62,25,52)(32,56,39,58,57,64,44,61);(1,5)(2,11)(3,16)(4,22)(6,26)(7,30)(8,27)(9,32)(10,36)(12,40)(13,42)(14,44)(15,45)(17,25)(18,48)(19,49)(20,51)(21,52)(23,53)(24,54)(28,41)(29,55)(31,56)(33,39)(34,57)(35,58)(37,59)(38,60)(43,61)(46,62)(47,63)(50,64);(4,23)(10,37)(15,46)(20,38)(21,47)(22,53)(24,29)(31,50)(35,43)(36,59)(45,62)(51,60)(52,63)(54,55)(56,64)(58,61)Polycyclic presentation:
2^9|a^2=i,c^2=g,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72287270, #72287292, #72287334, … |
| maximal normal subgroup of | #72287270, #72287292, #72287334, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6349897);