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, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1828 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)(10,12)(13,14,18,19)(15,22,25,20)(16,26,21,17)(23,27,28,24)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(6,9)(10,12)(14,19)(16,21)(20,22)(23,28)(29,30,33,36)(31,34,39,37)(32,35,41,40)(38,42,44,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,20,19,22)(16,23,21,28)(17,27,26,24)(29,30)(31,34)(32,40)(33,36)(35,41)(37,39)(38,43)(42,44);(1,2,3,4)(13,16,18,21)(14,17,19,26)(15,23,25,28)(20,27,22,24)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,7,8,11)(6,10,9,12)(13,17)(14,21)(15,24)(16,19)(18,26)(20,23)(22,28)(25,27)(29,32)(30,35)(31,38)(33,41)(34,42)(36,40)(37,43)(39,44)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=g,d^2=fgi,e^2=fgh,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, 4576790);