Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 | |
2703 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)(21,22,27,30)(23,33,35,44)(24,38,41,46)(25,32,26,31)(28,37,43,36)(29,39,45,40)(34,48,49,42)(47,50,52,51);(2,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23,27,35)(22,28,30,43)(24,34,41,49)(25,37,26,36)(29,42,45,48)(31,44,32,33)(38,52,46,47)(39,51,40,50);(6,10)(11,17)(12,18)(19,20)(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);(5,7,9,14)(6,11,10,17)(8,15,16,13)(12,20,18,19)(23,35)(28,43)(33,44)(34,49)(36,37)(42,48)(47,52)(50,51);(5,8,9,16)(6,12,10,18)(7,13,14,15)(11,19,17,20)(22,30)(28,43)(29,45)(31,32)(33,44)(38,46)(42,48)(47,52)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=i,d^2=h,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6442130);