Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60470 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#121 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 68 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 68 points:
(1,2,4,3)(5,6)(7,19)(8,25)(9,15)(10,16)(11,17)(12,18)(13,22)(14,27)(20,45)(21,44)(23,36)(24,35)(26,53)(28,38)(29,40)(30,41)(31,42)(32,39)(33,48)(34,50)(37,55)(43,47)(46,64)(49,60)(51,59)(52,58)(54,61)(56,62)(57,67)(63,66)(65,68);(5,7,11,23,12,24,10,22)(6,13,16,35,18,36,17,19)(8,20,32,52,28,48,27,47)(9,21,29,49,31,51,30,50)(14,33,38,58,39,45,25,43)(15,34,41,59,42,60,40,44)(26,46,54,65,56,67,55,66)(37,57,62,68,61,64,53,63);(5,8)(6,14)(7,20)(9,26)(10,27)(11,32)(12,28)(13,33)(15,37)(16,38)(17,25)(18,39)(19,43)(21,46)(22,47)(23,52)(24,48)(29,54)(30,55)(31,56)(34,57)(35,58)(36,45)(40,53)(41,62)(42,61)(44,63)(49,65)(50,66)(51,67)(59,68)(60,64);(1,3,4,2)(5,9,12,31)(6,15,18,42)(7,21,24,51)(8,26,28,56)(10,30,11,29)(13,34,36,60)(14,37,39,61)(16,41,17,40)(19,44,35,59)(20,46,48,67)(22,50,23,49)(25,53,38,62)(27,55,32,54)(33,57,45,64)(43,63,58,68)(47,66,52,65);(5,10,12,11)(6,16,18,17)(7,22,24,23)(8,27,28,32)(9,29,31,30)(13,35,36,19)(14,38,39,25)(15,40,42,41)(20,47,48,52)(21,49,51,50)(26,54,56,55)(33,58,45,43)(34,44,60,59)(37,53,61,62)(46,65,67,66)(57,63,64,68)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,g^c=gi,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280174);