Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5035983 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1627 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(5,6,9,12)(7,13,15,10)(8,11,16,18)(14,19,20,17)(21,22,25,26)(23,27,31,29)(24,28,33,34)(30,35,32,36)(38,42)(40,48)(43,45)(46,49);(8,16)(11,18)(14,20)(17,19)(22,26)(27,29)(28,34)(35,36)(37,38,41,42)(39,45,47,43)(40,44,48,50)(46,52,49,51);(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23)(22,27)(24,32)(25,31)(26,29)(28,36)(30,33)(34,35)(37,39,41,47)(38,43,42,45)(40,49,48,46)(44,52,50,51);(1,2,3,4)(5,7,9,15)(6,10,12,13)(8,14,16,20)(11,17,18,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,36)(32,35)(33,34)(37,38)(39,45)(40,44)(41,42)(43,47)(46,52)(48,50)(49,51);(5,8)(6,11)(7,14)(9,16)(10,17)(12,18)(13,19)(15,20)(21,24,25,33)(22,28,26,34)(23,30,31,32)(27,35,29,36)(37,40,41,48)(38,44,42,50)(39,46,47,49)(43,51,45,52)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=h,d^2=fi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4989592);