Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #222240 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D4×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2167 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 68 points:
(5,6,12,16,11,17,10,30)(7,18,21,13,22,37,31,36)(8,23,25,40,26,14,33,41)(9,27,28,15,29,43,32,42)(19,44,49,38,48,62,47,34)(20,35,52,60,51,61,50,45)(24,54,57,39,56,58,55,53)(46,59,65,64,66,63,67,68);(1,2,3,4)(5,7,10,31,11,22,12,21)(6,13,30,18,17,36,16,37)(8,19,33,47,26,48,25,49)(9,20,32,50,29,51,28,52)(14,38,40,44,23,34,41,62)(15,35,27,45,42,61,43,60)(24,46,55,67,56,66,57,65)(39,63,54,64,53,59,58,68);(2,4)(5,8)(6,14)(7,19)(9,24)(10,25)(11,26)(12,33)(13,34)(15,39)(16,40)(17,23)(18,44)(20,46)(21,47)(22,48)(27,58)(28,55)(29,56)(30,41)(31,49)(32,57)(35,59)(36,38)(37,62)(42,53)(43,54)(45,64)(50,65)(51,66)(52,67)(60,68)(61,63);(5,9,11,29)(6,15,17,42)(7,20,22,51)(8,24,26,56)(10,32,12,28)(13,35,36,61)(14,39,23,53)(16,43,30,27)(18,45,37,60)(19,46,48,66)(21,52,31,50)(25,57,33,55)(34,59,38,63)(40,54,41,58)(44,64,62,68)(47,67,49,65)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,d^2=i,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273922);