Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1598 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(6,9)(10,12)(13,14,18,17)(15,21,23,19)(16,20,24,26)(22,28,25,27)(29,30,33,36)(31,37,39,34)(32,35,40,42)(38,43,44,41);(1,2,3,4)(14,17)(16,24)(19,21)(22,25)(32,40)(35,42)(38,44)(41,43);(1,2)(3,4)(13,15)(14,19)(16,25)(17,21)(18,23)(20,28)(22,24)(26,27)(29,31,33,39)(30,34,36,37)(32,38,40,44)(35,41,42,43);(5,6,8,9)(7,10,11,12)(13,16)(14,20)(15,22)(17,26)(18,24)(19,27)(21,28)(23,25)(29,32,33,40)(30,35,36,42)(31,38,39,44)(34,41,37,43);(5,7,8,11)(6,10,9,12)(13,17,18,14)(15,21,23,19)(16,20,24,26)(22,27,25,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,41)(40,42)(43,44)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fg,d^2=fgi,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3176869);