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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1748 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)(5,6,10,8)(7,12,9,11)(15,21)(16,23)(19,24)(20,25)(29,30,34,37)(31,38,32,35)(33,36,41,44)(39,43,40,42);(5,6)(7,11)(8,10)(9,12)(14,18)(19,24)(20,25)(26,28)(29,31,34,32)(30,35,37,38)(33,40,41,39)(36,43,44,42);(5,7,10,9)(6,11,8,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,32,34,31)(30,35,37,38)(33,40,41,39)(36,42,44,43);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,30)(31,35)(32,38)(33,36)(34,37)(39,42)(40,43)(41,44);(5,9,10,7)(6,11,8,12)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,41)(30,36,37,44)(31,39,32,40)(35,42,38,43)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=fh,d^2=gh,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, 4376427);