Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6443343 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2433 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4838 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)(3,15)(4,21)(5,11)(6,13)(7,12)(8,14)(9,18)(10,25)(16,31)(17,42)(19,34)(20,33)(22,50)(23,49)(24,39)(26,55)(27,40)(28,38)(29,36)(30,37)(32,47)(35,54)(41,44)(43,56)(45,57)(46,48)(51,64)(52,59)(53,58)(60,62)(61,63);(1,3,6,18,8,20,7,19)(2,9,12,15,14,34,13,33)(4,22,25,46,24,16,29,44)(5,26,27,47,28,17,30,45)(10,31,21,48,36,50,39,41)(11,32,37,55,38,57,40,42)(23,43,53,61,52,51,54,62)(35,56,59,60,58,64,49,63);(1,4,7,25,8,24,6,29)(2,10,13,21,14,36,12,39)(3,16,19,44,20,22,18,46)(5,23,27,54,28,52,30,53)(9,31,33,48,34,50,15,41)(11,35,37,49,38,58,40,59)(17,43,45,62,26,51,47,61)(32,56,55,63,57,64,42,60);(1,5)(2,11)(3,17)(4,23)(6,30)(7,27)(8,28)(9,32)(10,35)(12,40)(13,37)(14,38)(15,42)(16,43)(18,47)(19,45)(20,26)(21,49)(22,51)(24,52)(25,54)(29,53)(31,56)(33,55)(34,57)(36,58)(39,59)(41,60)(44,62)(46,61)(48,63)(50,64);(4,24)(10,36)(16,22)(21,39)(23,52)(25,29)(31,50)(35,58)(41,48)(43,51)(44,46)(49,59)(53,54)(56,64)(60,63)(61,62)Polycyclic presentation:
2^9|b^2=f,c^2=g,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,h^b=hi,e^c=ei,f^c=fi,f^d=fi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72303765, #72303769, #72303792, … |
| maximal normal subgroup of | #72303765, #72303769, #72303792, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350539);