Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500186 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1306 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)(4,8)(5,7)(25,26,31,44,30,34,29,35)(27,36,43,32,37,49,39,46)(28,40,45,48,41,33,42,55)(38,52,56,54,51,50,53,47);(1,3,5,4)(2,6,7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,27,30,37)(26,32,34,46)(28,38,41,51)(29,43,31,39)(33,47,40,54)(35,49,44,36)(42,56,45,53)(48,52,55,50);(2,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(27,37)(28,41)(32,46)(33,40)(36,49)(39,43)(42,45)(48,55);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,28)(26,33)(27,38)(29,42)(30,41)(31,45)(32,47)(34,40)(35,48)(36,50)(37,51)(39,53)(43,56)(44,55)(46,54)(49,52);(1,4,5,3)(2,8,7,6)(10,16)(14,17)(15,24)(22,23)(26,34)(27,37)(28,41)(35,44)(39,43)(42,45)(47,54)(50,52)Polycyclic presentation:
2^9|a^2=fi,b^2=gi,c^2=hi,d^2=hi,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6439701);