Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2794 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)(23,33)(24,38)(25,30)(26,31)(27,32)(28,36)(29,40)(34,48)(35,44)(37,43)(39,46)(41,45)(42,49)(47,50)(51,52);(2,4)(5,6)(7,11)(8,12)(9,10)(13,19)(14,17)(15,20)(16,18)(21,23)(22,28)(24,34)(25,36)(26,37)(27,35)(29,42)(30,44)(31,33)(32,43)(38,52)(39,50)(40,51)(41,49)(45,47)(46,48);(6,10)(11,17)(12,18)(19,20)(21,24,27,41)(22,29,32,46)(23,34,35,49)(25,39,26,40)(28,42,43,48)(30,38,31,45)(33,47,44,52)(36,50,37,51);(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)(21,25,27,26)(22,30,32,31)(23,36,35,37)(24,39,41,40)(28,44,43,33)(29,38,46,45)(34,50,49,51)(42,52,48,47)Polycyclic presentation:
2^9|a^2=fg,c^2=i,d^2=h,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,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=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451406);