Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| 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×#3552 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2860 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)(5,6)(7,10)(8,12)(9,11)(13,17)(14,16)(15,18)(19,20)(21,22,26,32,27,30,25,31)(23,33,37,28,35,44,36,43)(24,38,40,45,41,46,39,29)(34,48,51,52,49,42,50,47);(2,4)(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23)(22,28)(24,34)(25,37)(26,36)(27,35)(29,42)(30,43)(31,44)(32,33)(38,52)(39,51)(40,50)(41,49)(45,48)(46,47);(6,11)(7,13)(8,15)(16,20)(21,24,27,41)(22,29,30,45)(23,34,35,49)(25,40,26,39)(28,42,43,48)(31,38,32,46)(33,47,44,52)(36,51,37,50);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,16,17,20)(22,30)(23,35)(29,45)(31,32)(34,49)(36,37)(38,46)(50,51);(7,13)(10,17)(14,19)(16,20)(22,30)(28,43)(29,45)(31,32)(33,44)(38,46)(42,48)(47,52)Polycyclic presentation:
2^9|a^2=gh,c^2=i,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6431994);