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 | |
1616 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)(5,6,10,11)(7,12,9,8)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30)(31,36)(32,37)(33,38)(34,35)(39,44)(40,41)(42,43);(5,7,10,9)(6,8,11,12)(15,21)(19,22)(23,27)(26,28)(30,35)(31,37)(38,43)(39,41);(6,11)(7,9)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,31,34,37)(30,36,35,32)(33,41,42,39)(38,40,43,44);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,32,34,36)(30,37,35,31)(33,40,42,44)(38,41,43,39);(5,9,10,7)(6,8,11,12)(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,33)(30,38)(31,39)(32,40)(34,42)(35,43)(36,44)(37,41)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=fg,d^2=gh,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, 4513489);