Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
1719 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,26,30)(23,27,32,41)(24,28,36,45)(25,29,39,48)(31,40,49,51)(33,42,34,43)(35,44,50,52)(37,46,38,47);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23)(22,27)(24,35)(25,33)(26,32)(28,44)(29,42)(30,41)(31,37)(34,39)(36,50)(38,49)(40,46)(43,48)(45,52)(47,51);(1,2,3,4)(21,24)(22,28)(23,31)(25,38)(26,36)(27,40)(29,47)(30,45)(32,49)(33,35)(34,50)(37,39)(41,51)(42,44)(43,52)(46,48);(6,11)(10,13)(12,19)(18,20)(23,32)(24,36)(27,41)(28,45)(33,34)(37,38)(42,43)(46,47);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(23,33)(25,39)(27,42)(29,48)(31,35)(32,34)(37,38)(40,44)(41,43)(46,47)(49,50)(51,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,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, 6593440);