Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6,8,10)(7,11,12,9)(14,18)(15,21)(20,25)(22,27)(29,30,34,35)(31,37,38,33)(32,40,41,36)(39,42,44,43);(5,7,8,12)(6,9,10,11)(15,21)(19,24)(22,27)(26,28)(30,35)(31,38)(32,41)(42,43);(6,10)(9,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,38)(30,33,35,37)(32,39,41,44)(36,42,40,43);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32,34,41)(30,36,35,40)(31,39,38,44)(33,42,37,43);(5,6)(7,9)(8,10)(11,12)(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,33)(30,31)(32,42)(34,37)(35,38)(36,39)(40,44)(41,43)Polycyclic presentation:
2^9|a^2=f,b^2=fh,c^2=fgh,d^2=fhi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4575675);