Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420420 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60645 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1854 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(1,2)(3,4)(5,6)(7,19)(8,24)(9,15)(10,17)(11,16)(12,18)(13,22)(14,28)(20,48)(21,47)(23,45)(25,35)(26,54)(27,40)(29,39)(30,61)(31,44)(32,43)(33,42)(34,37)(36,51)(38,58)(41,52)(46,50)(49,68)(53,56)(55,63)(57,67)(59,66)(60,65)(62,64);(5,7,10,22,12,23,11,34)(6,13,16,19,18,37,17,45)(8,25,27,56,29,20,28,50)(9,30,31,52,32,51,33,21)(14,35,39,46,40,48,24,53)(15,41,42,61,43,47,44,36)(26,57,58,49,59,62,60,55)(38,64,54,68,65,67,66,63);(5,8,11,28,12,29,10,27)(6,14,17,24,18,40,16,39)(7,20,22,50,23,25,34,56)(9,26,31,58,32,59,33,60)(13,35,19,46,37,48,45,53)(15,38,42,54,43,65,44,66)(21,49,51,57,52,55,30,62)(36,63,47,67,61,68,41,64);(2,4)(5,9,12,32)(6,15,18,43)(7,21,23,52)(8,26,29,59)(10,33,11,31)(13,36,37,61)(14,38,40,65)(16,44,17,42)(19,47,45,41)(20,49,25,55)(22,51,34,30)(24,54,39,66)(27,60,28,58)(35,63,48,68)(46,67,53,64)(50,57,56,62)Polycyclic presentation:
2^9|b^2=e,c^2=f,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330030);