Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2)(3,4)(21,22)(23,32)(24,29)(25,30)(26,41)(27,31)(28,35)(33,49)(34,43)(36,48)(37,52)(38,45)(39,46)(40,47)(42,50)(44,51);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,27,36)(22,28,31,43)(24,37,38,33)(25,35,26,34)(29,44,45,42)(30,48,41,32)(39,50,40,51)(46,52,47,49);(2,4)(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,27,38)(22,29,31,45)(23,33,36,37)(25,40,26,39)(28,42,43,44)(30,47,41,46)(32,49,48,52)(34,51,35,50);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(24,38)(29,45)(33,37)(39,40)(42,44)(46,47)(49,52)(50,51);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,31,41)(23,34,36,35)(24,39,38,40)(28,32,43,48)(29,46,45,47)(33,50,37,51)(42,52,44,49)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436171);