Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| 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 | |
1538 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)(5,6,10,7)(8,13,16,11)(9,14,17,12)(15,18,20,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36)(38,42)(39,45)(40,48)(50,52);(6,7)(9,17)(11,13)(15,20)(22,26)(24,32)(27,29)(30,36)(37,38,41,42)(39,46,45,43)(40,49,48,44)(47,50,51,52);(5,7,10,6)(8,11,16,13)(9,14,17,12)(15,18,20,19)(21,23,25,31)(22,27,26,29)(24,30,32,36)(28,34,33,35)(37,38)(39,46)(40,44)(41,42)(43,45)(47,52)(48,49)(50,51);(1,2)(3,4)(5,8,10,16)(6,11,7,13)(9,15,17,20)(12,18,14,19)(21,22)(23,29)(24,28)(25,26)(27,31)(30,35)(32,33)(34,36)(37,39,41,45)(38,43,42,46)(40,47,48,51)(44,50,49,52);(5,9)(6,12)(7,14)(8,15)(10,17)(11,18)(13,19)(16,20)(21,24,25,32)(22,28,26,33)(23,30,31,36)(27,34,29,35)(37,40,41,48)(38,44,42,49)(39,47,45,51)(43,50,46,52)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,c^2=gh,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4377263);