Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1542 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)(6,10)(7,13)(12,17)(14,19)(21,22,25,26)(23,29,30,27)(24,28,33,32)(31,35,36,34)(37,38,43,46)(39,47,40,44)(41,42,52,45)(48,51,50,49);(7,13)(11,16)(14,19)(18,20)(22,26)(23,30)(28,32)(31,36)(37,39,43,40)(38,44,46,47)(41,48,52,50)(42,49,45,51);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,22)(23,27)(24,32)(25,26)(28,33)(29,30)(31,35)(34,36)(37,40,43,39)(38,44,46,47)(41,50,52,48)(42,49,45,51);(1,2,3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23,25,30)(22,27,26,29)(24,31,33,36)(28,34,32,35)(37,41)(38,45)(39,48)(40,50)(42,46)(43,52)(44,51)(47,49);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24)(22,28)(23,31)(25,33)(26,32)(27,34)(29,35)(30,36)(37,42,43,45)(38,41,46,52)(39,49,40,51)(44,48,47,50)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fh,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4575547);