Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1710 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)(5,6,8,10)(7,9,12,11)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(29,30,34,38)(31,36,32,35)(33,41,42,37)(39,43,40,44);(5,6)(7,9)(8,10)(11,12)(15,21)(19,22)(23,27)(26,28)(29,31,34,32)(30,35,38,36)(33,40,42,39)(37,44,41,43);(5,6,8,10)(7,9,12,11)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27)(29,32,34,31)(30,36,38,35)(33,40,42,39)(37,44,41,43);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,30)(31,35)(32,36)(33,41)(34,38)(37,42)(39,44)(40,43);(5,7)(6,9)(8,12)(10,11)(13,17,18,25)(14,20,16,24)(15,23,21,27)(19,26,22,28)(29,33)(30,37)(31,39)(32,40)(34,42)(35,43)(36,44)(38,41)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=f,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5539423);