Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6131635 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1590 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(7,13)(8,15)(10,16)(11,18)(22,26)(23,29)(28,33)(30,35)(37,38,41,44)(39,42,45,48)(40,43,47,50)(46,49,51,52);(1,2,3,4)(7,13)(10,16)(14,19)(17,20)(22,26)(24,31)(27,32)(30,35)(39,45)(40,47)(42,48)(43,50);(5,6,9,12)(7,10,13,16)(8,11,15,18)(14,17,19,20)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36)(37,38)(39,42)(40,43)(41,44)(45,48)(46,49)(47,50)(51,52);(5,7)(6,10)(8,14)(9,13)(11,17)(12,16)(15,19)(18,20)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36)(37,39,41,45)(38,42,44,48)(40,46,47,51)(43,49,50,52);(5,8)(6,11)(7,14)(9,15)(10,17)(12,18)(13,19)(16,20)(21,24,25,31)(22,28,26,33)(23,30,29,35)(27,34,32,36)(37,40)(38,43)(39,46)(41,47)(42,49)(44,50)(45,51)(48,52)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=i,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=eg68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6039817);