Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1628 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30,34,31)(32,37,39,35)(33,36,42,38)(40,44,41,43);(5,6,9,8)(7,11,12,10)(15,22)(18,21)(23,28)(26,27)(30,31)(32,39)(36,38)(40,41);(5,7,9,12)(6,10,8,11)(13,14)(15,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28)(29,31,34,30)(32,37,39,35)(33,36,42,38)(40,43,41,44);(1,2)(3,4)(5,6,9,8)(7,10,12,11)(13,15,17,22)(14,18,20,21)(16,23,25,28)(19,26,24,27)(29,32)(30,35)(31,37)(33,41)(34,39)(36,44)(38,43)(40,42);(5,8,9,6)(7,11,12,10)(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33)(30,36)(31,38)(32,40)(34,42)(35,43)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=fg,b^2=fh,c^2=h,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3271460);