Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417700 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60368 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2309 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)(8,25)(9,15)(10,17)(11,16)(12,18)(13,22)(14,27)(20,38)(21,47)(23,37)(24,36)(26,54)(28,45)(29,59)(30,44)(31,43)(32,42)(33,40)(34,50)(35,52)(39,57)(41,51)(46,49)(48,62)(53,61)(55,66)(56,65)(58,64)(60,68)(63,67);(5,7,12,24)(6,13,18,37)(8,20,28,50)(9,29,31,52)(10,23,11,22)(14,38,40,34)(15,41,43,47)(16,36,17,19)(21,32,51,30)(25,46,45,61)(26,55,58,67)(27,53,33,49)(35,44,59,42)(39,63,65,66)(48,57,68,56)(54,60,64,62);(5,8)(6,14)(7,20)(9,26)(10,27)(11,33)(12,28)(13,34)(15,39)(16,25)(17,45)(18,40)(19,46)(21,48)(22,49)(23,53)(24,50)(29,55)(30,56)(31,58)(32,57)(35,60)(36,61)(37,38)(41,66)(42,64)(43,65)(44,54)(47,63)(51,68)(52,67)(59,62);(2,4)(5,9,12,31)(6,15,18,43)(7,21,24,51)(8,26,28,58)(10,32,11,30)(13,35,37,59)(14,39,40,65)(16,44,17,42)(19,47,36,41)(20,48,50,68)(22,52,23,29)(25,54,45,64)(27,57,33,56)(34,60,38,62)(46,63,61,66)(49,67,53,55)Polycyclic presentation:
2^9|b^2=i,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,e^b=ei,f^b=fi,f^c=fi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 324913);