Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #220200 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1600 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)(7,17)(8,22,26,13)(9,27)(10,31)(11,16)(12,20)(14,29)(15,28)(18,38,23,50)(19,43)(21,40)(24,54,56,57)(25,36,51,49)(30,58)(32,53,47,42)(33,46)(34,44)(35,55,48,66)(37,59)(39,45)(41,63,52,60)(61,68,67,64)(62,65);(5,7,10,20,11,21,28,44)(6,12,15,17,16,34,31,40)(8,23,51,42,26,18,25,53)(9,19,29,45,30,46,59,65)(13,32,49,50,22,47,36,38)(14,33,27,39,37,43,58,62)(24,52,66,64,56,41,55,68)(35,60,54,67,48,63,57,61);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,32)(14,35)(15,36)(16,22)(17,38)(19,41)(20,42)(21,23)(27,57)(28,51)(29,55)(30,56)(31,49)(33,60)(34,47)(37,48)(39,61)(40,50)(43,63)(44,53)(45,64)(46,52)(54,58)(59,66)(62,67)(65,68);(5,9,28,59,11,30,10,29)(6,14,31,58,16,37,15,27)(7,19,44,65,21,46,20,45)(8,24,51,66,26,56,25,55)(12,33,40,62,34,43,17,39)(13,35,49,54,22,48,36,57)(18,41,53,68,23,52,42,64)(32,60,50,67,47,63,38,61)Polycyclic presentation:
2^9|a^2=h,b^2=eh,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^c=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272268);