Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1864 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)(9,12)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26)(29,30)(31,36)(32,33)(34,35)(37,42)(38,44)(39,41)(40,43);(5,6,8,10)(7,9,11,12)(14,18)(16,24)(19,21)(22,28)(30,35)(33,37)(36,43)(39,44);(5,6)(7,9)(8,10)(11,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,31)(30,36)(32,41)(33,39)(34,40)(35,43)(37,44)(38,42);(1,2)(3,4)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28)(29,32)(30,37)(31,38)(33,35)(34,42)(36,39)(40,41)(43,44);(5,7)(6,9)(8,11)(10,12)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)(29,33,34,37)(30,32,35,42)(31,39,40,44)(36,41,43,38)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=f,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, 4373608);