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, cosocle | C24 | C25 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14,17,20)(15,21,23,18)(16,19,24,26)(22,27,28,25)(29,30)(31,37)(32,34)(33,40)(35,39)(36,41)(38,43)(42,44);(6,9)(10,12)(16,24)(19,26)(22,28)(25,27)(30,32)(33,41)(35,37)(38,42);(5,6)(7,10)(8,9)(11,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28)(29,31)(30,35)(32,37)(33,42)(34,39)(36,44)(38,41)(40,43);(1,2,3,4)(13,15,17,23)(14,18,20,21)(16,22,24,28)(19,25,26,27)(29,32,34,30)(31,37,39,35)(33,40,41,36)(38,44,42,43);(5,7,8,11)(6,10,9,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)(29,33)(30,36)(31,38)(32,40)(34,41)(35,43)(37,44)(39,42)Polycyclic presentation:
2^9|a^2=fg,d^2=i,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, 5189004);