Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #392 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,30,33)(27,31,38,45)(28,32,40,50)(29,34,43,51)(35,44,53,55)(36,46,37,47)(39,48,54,56)(41,49,42,52);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,27)(26,31)(28,39)(29,36)(30,38)(32,48)(33,45)(34,46)(35,41)(37,43)(40,54)(42,53)(44,49)(47,51)(50,56)(52,55);(2,5)(6,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,32)(27,35)(29,42)(30,40)(31,44)(33,50)(34,52)(36,39)(37,54)(38,53)(41,43)(45,55)(46,48)(47,56)(49,51);(10,16)(14,17)(15,22)(21,23)(26,33)(28,40)(31,45)(34,51)(35,53)(39,54)(41,42)(46,47);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(21,23)(27,36)(29,43)(31,46)(34,51)(35,39)(37,38)(41,42)(44,48)(45,47)(49,52)(53,54)(55,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6619408);