Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4131229 |
| 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 | |
1718 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)(7,11)(10,12)(13,14,18,19)(15,20,22,16)(17,21,27,28)(23,26,25,24)(29,30)(31,36)(32,38)(33,35)(34,41)(37,42)(39,44)(40,43);(6,9)(10,12)(14,19)(15,22)(21,28)(23,25)(29,31,35,32)(30,36,33,38)(34,40,42,39)(37,44,41,43);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,19)(17,25)(18,22)(21,24)(23,27)(26,28)(29,32,35,31)(30,36,33,38)(34,40,42,39)(37,43,41,44);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16,18,20)(14,15,19,22)(17,26,27,24)(21,25,28,23)(29,33,35,30)(31,38,32,36)(34,41,42,37)(39,44,40,43);(13,17,18,27)(14,21,19,28)(15,23,22,25)(16,24,20,26)(29,34)(30,37)(31,39)(32,40)(33,41)(35,42)(36,43)(38,44)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fgh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4049678);