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 | |
1902 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)(15,20,22,16)(17,21,26,27)(23,28,25,24)(29,30,34,36)(31,37,39,32)(33,35,42,44)(38,43,41,40);(6,9)(10,12)(14,19)(15,22)(21,27)(23,25)(29,31,34,39)(30,32,36,37)(33,38,42,41)(35,40,44,43);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,19)(17,25)(18,22)(21,24)(23,26)(27,28)(29,31)(30,32)(33,41)(34,39)(35,43)(36,37)(38,42)(40,44);(1,2)(3,4)(13,16,18,20)(14,15,19,22)(17,24,26,28)(21,23,27,25)(29,32)(30,31)(33,40)(34,37)(35,38)(36,39)(41,44)(42,43);(5,7)(6,10)(8,11)(9,12)(13,17)(14,21)(15,23)(16,24)(18,26)(19,27)(20,28)(22,25)(29,33)(30,35)(31,38)(32,40)(34,42)(36,44)(37,43)(39,41)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fh,d^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, 4506390);