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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1648 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)(6,10)(7,13)(12,17)(14,19)(22,26)(23,29)(24,31)(34,36)(37,38)(39,45)(40,44)(41,42)(43,47)(46,51)(48,49)(50,52);(7,13)(11,16)(14,19)(18,20)(22,26)(23,29)(28,33)(30,35)(38,42)(43,45)(44,48)(50,51);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,22,25,26)(23,27,29,32)(24,28,31,33)(30,34,35,36)(37,38)(39,43)(40,48)(41,42)(44,49)(45,47)(46,51)(50,52);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,23)(22,27)(24,30)(25,29)(26,32)(28,34)(31,35)(33,36)(37,39)(38,43)(40,46)(41,47)(42,45)(44,50)(48,51)(49,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,24)(22,28)(23,30)(25,31)(26,33)(27,34)(29,35)(32,36)(37,40,41,49)(38,44,42,48)(39,46,47,52)(43,50,45,51)Polycyclic presentation:
2^9|a^2=fgi,c^2=g,d^2=fgh,e^2=gh,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, 4520837);