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 | |
1528 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)(5,6,9,10)(7,13,14,11)(8,12,16,18)(15,20,17,19)(21,22,25,28)(23,26,29,33)(24,31,32,27)(30,36,35,34)(39,45)(41,49)(43,46)(44,48);(6,10)(7,14)(8,16)(19,20)(23,29)(26,33)(30,35)(34,36)(37,38,42,40)(39,46,45,43)(41,48,49,44)(47,50,51,52);(5,7,9,14)(6,11,10,13)(8,17,16,15)(12,20,18,19)(21,23,25,29)(22,26,28,33)(24,30,32,35)(27,34,31,36)(37,39,42,45)(38,43,40,46)(41,47,49,51)(44,50,48,52);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,14)(15,19)(16,18)(17,20)(21,22)(23,26)(24,27)(25,28)(29,33)(30,34)(31,32)(35,36)(37,40,42,38)(39,43,45,46)(41,48,49,44)(47,50,51,52);(5,8,9,16)(6,12,10,18)(7,15,14,17)(11,19,13,20)(21,24,25,32)(22,27,28,31)(23,30,29,35)(26,34,33,36)(37,41)(38,44)(39,47)(40,48)(42,49)(43,50)(45,51)(46,52)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=h,d^2=fg,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, 5130822);