Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1778 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)(15,21,22,17)(16,24,25,20)(23,26,28,27)(29,30,34,37)(31,32,39,35)(33,36,42,44)(38,40,41,43);(6,9)(10,12)(14,19)(15,22)(16,25)(26,27)(29,31,34,39)(30,35,37,32)(33,38,42,41)(36,43,44,40);(5,6)(7,10)(8,9)(11,12)(13,15,18,22)(14,17,19,21)(16,23,25,28)(20,26,24,27)(29,31)(30,32)(33,41)(34,39)(35,37)(36,43)(38,42)(40,44);(1,2,3,4)(13,16,18,25)(14,20,19,24)(15,23,22,28)(17,26,21,27)(29,32)(30,31)(33,40)(34,35)(36,38)(37,39)(41,44)(42,43);(5,7)(6,10)(8,11)(9,12)(13,17)(14,15)(16,26)(18,21)(19,22)(20,23)(24,28)(25,27)(29,33)(30,36)(31,38)(32,40)(34,42)(35,43)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=f,b^2=fg,c^2=g,d^2=fgi,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, 4571869);