Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499804 |
| 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×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2740 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)(23,28)(24,35)(25,31)(26,40)(27,30)(29,38)(32,49)(33,45)(34,43)(36,52)(37,47)(39,44)(41,48)(42,51)(46,50);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,27,33)(22,28,30,45)(24,36,37,32)(25,39,26,34)(29,46,48,42)(31,44,40,43)(35,52,47,49)(38,50,41,51);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,27,37)(22,29,30,48)(23,32,33,36)(25,38,26,41)(28,42,45,46)(31,47,40,35)(34,50,39,51)(43,49,44,52);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,30)(23,33)(24,37)(31,40)(34,39)(38,41)(42,46)(49,52);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,31,30,40)(23,34,33,39)(24,38,37,41)(28,43,45,44)(29,47,48,35)(32,50,36,51)(42,49,46,52)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436445);