Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1544 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)(5,6)(7,13)(8,12)(9,10)(11,14)(15,20)(16,18)(17,19)(21,22,26,24)(23,27,29,30)(25,32,33,28)(31,36,35,34)(37,38,42,46)(39,40,44,43)(41,45,50,52)(47,48,49,51);(6,10)(7,14)(8,16)(19,20)(23,29)(27,30)(31,35)(34,36)(37,39,42,44)(38,43,46,40)(41,47,50,49)(45,51,52,48);(5,7)(6,11)(8,17)(9,14)(10,13)(12,20)(15,16)(18,19)(21,23,26,29)(22,27,24,30)(25,31,33,35)(28,34,32,36)(37,39)(38,40)(41,49)(42,44)(43,46)(45,51)(47,50)(48,52);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,18)(17,20)(21,24,26,22)(23,30,29,27)(25,32,33,28)(31,36,35,34)(37,40)(38,44)(39,46)(41,48)(42,43)(45,49)(47,52)(50,51);(5,8,9,16)(6,12,10,18)(7,15,14,17)(11,19,13,20)(21,25,26,33)(22,28,24,32)(23,31,29,35)(27,34,30,36)(37,41)(38,45)(39,47)(40,48)(42,50)(43,51)(44,49)(46,52)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fh,d^2=fh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,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, 5134665);