Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6439025 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4814 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(2,4)(5,6,12,18)(7,19,22,38)(8,25,30,41)(9,15,34,46)(10,17,11,16)(13,24,37,23)(14,28,42,29)(20,47,49,35)(21,43,51,61)(26,39,50,59)(27,53,57,63)(31,36,52,60)(32,45,33,44)(40,56,64,55)(48,65,66,58)(54,62,67,68);(1,2,3,4)(5,7,10,23,12,22,11,24)(6,13,16,19,18,37,17,38)(8,26,28,20,30,50,29,49)(9,31,32,21,34,52,33,51)(14,39,41,35,42,59,25,47)(15,43,44,36,46,61,45,60)(27,48,55,54,57,66,56,67)(40,58,53,62,64,65,63,68);(5,8,11,28,12,30,10,29)(6,14,17,41,18,42,16,25)(7,20,24,50,22,49,23,26)(9,27,32,56,34,57,33,55)(13,35,38,59,37,47,19,39)(15,40,44,63,46,64,45,53)(21,48,52,67,51,66,31,54)(36,58,61,68,60,65,43,62);(5,9)(6,15)(7,21)(8,27)(10,33)(11,32)(12,34)(13,36)(14,40)(16,45)(17,44)(18,46)(19,43)(20,48)(22,51)(23,31)(24,52)(25,53)(26,54)(28,56)(29,55)(30,57)(35,58)(37,60)(38,61)(39,62)(41,63)(42,64)(47,65)(49,66)(50,67)(59,68);(7,22)(13,37)(19,38)(20,49)(21,51)(23,24)(26,50)(31,52)(35,47)(36,60)(39,59)(43,61)(48,66)(54,67)(58,65)(62,68)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6350484);