Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
1752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,10,8)(7,11,9,12)(14,18)(16,23)(19,24)(22,27)(29,30)(31,35)(32,40)(33,34)(36,42)(37,38)(39,44)(41,43);(2,4)(5,7,10,9)(6,11,8,12)(14,18)(15,21)(20,25)(22,27)(30,34)(31,37)(36,40)(39,41);(1,2,3,4)(5,8,10,6)(7,11,9,12)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,30)(31,38)(32,36)(33,34)(35,37)(39,44)(40,42)(41,43);(5,9,10,7)(6,11,8,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,31)(30,35)(32,41)(33,37)(34,38)(36,44)(39,42)(40,43);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32,33,42)(30,36,34,40)(31,39,37,41)(35,43,38,44)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fh,d^2=gi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2687165);