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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1785 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)(13,14,18,19)(15,20,23,27)(16,24,25,17)(21,22,28,26)(29,30)(31,38)(32,35)(33,36)(34,40)(37,42)(39,43)(41,44);(5,6,8,10)(7,9,11,12)(14,19)(16,25)(20,27)(21,28)(29,31,35,33)(30,36,32,38)(34,39,42,41)(37,43,40,44);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,21)(17,26)(18,23)(19,27)(22,24)(25,28)(29,32,35,30)(31,36,33,38)(34,37,42,40)(39,44,41,43);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,16,18,25)(14,17,19,24)(15,21,23,28)(20,26,27,22)(29,33,35,31)(30,36,32,38)(34,39,42,41)(37,44,40,43);(13,17)(14,16)(15,22)(18,24)(19,25)(20,28)(21,27)(23,26)(29,34,35,42)(30,37,32,40)(31,39,33,41)(36,43,38,44)Polycyclic presentation:
2^9|a^2=fg,b^2=h,c^2=g,d^2=hi,e^2=g,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, 3376906);