Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6444406 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #60450 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2614 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(5,6,12,17,11,18,10,16)(7,19,23,13,24,46,22,37)(8,25,33,40,28,41,29,14)(9,15,34,45,31,43,32,44)(20,47,50,38,49,35,26,54)(21,42,52,36,51,66,30,59)(27,53,60,65,58,64,57,39)(48,63,67,62,56,61,55,68);(1,2)(3,4)(5,7)(6,13)(8,26)(9,30)(10,23)(11,24)(12,22)(14,38)(15,42)(16,46)(17,19)(18,37)(20,29)(21,32)(25,47)(27,56)(28,50)(31,52)(33,49)(34,51)(35,41)(36,44)(39,63)(40,54)(43,66)(45,59)(48,58)(53,68)(55,57)(60,67)(61,65)(62,64);(2,4)(5,8)(6,14,18,40)(7,20,24,49)(9,27)(10,29)(11,28)(12,33)(13,35)(15,39,43,65)(16,41,17,25)(19,38)(21,48,51,56)(22,26,23,50)(30,55,52,67)(31,58)(32,57)(34,60)(36,61)(37,47)(42,62)(44,64,45,53)(46,54)(59,63)(66,68);(2,4)(5,9,10,32,11,31,12,34)(6,15,16,44,18,43,17,45)(7,21,22,30,24,51,23,52)(8,27,29,57,28,58,33,60)(13,36,19,42,37,59,46,66)(14,39,41,64,40,65,25,53)(20,48,26,55,49,56,50,67)(35,61,38,62,47,63,54,68);(8,28)(9,31)(14,40)(15,43)(20,49)(21,51)(25,41)(26,50)(29,33)(30,52)(32,34)(35,47)(36,59)(38,54)(42,66)(44,45)Polycyclic presentation:
2^9|a^2=fi,c^2=fgi,d^2=f,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351608);