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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1911 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)(7,11)(8,12)(9,10)(13,14,17,18)(15,21,23,19)(16,24,26,20)(22,27,25,28)(29,30)(31,37)(32,34)(33,36)(35,38)(39,44)(40,42)(41,43);(5,7,10,8)(6,11,9,12)(14,18)(19,21)(20,24)(27,28)(30,32)(31,38)(36,40)(39,41);(5,8,10,7)(6,11,9,12)(13,15)(14,19)(16,25)(17,23)(18,21)(20,28)(22,26)(24,27)(29,31)(30,35)(32,37)(33,41)(34,38)(36,44)(39,42)(40,43);(1,2,3,4)(5,9,10,6)(7,11,8,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,26)(25,28)(29,32,34,30)(31,37,38,35)(33,40,42,36)(39,44,41,43);(13,16)(14,20)(15,22)(17,26)(18,24)(19,27)(21,28)(23,25)(29,33,34,42)(30,36,32,40)(31,39,38,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fg,d^2=fhi,e^2=h,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, 4579051);