Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| 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 | |
1816 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)(13,14,18,20)(15,21,22,16)(17,25,26,19)(23,24,28,27)(29,30,34,38)(31,36,32,35)(33,37,42,41)(39,44,40,43);(6,9)(10,12)(15,22)(16,21)(23,28)(24,27)(29,31,34,32)(30,35,38,36)(33,40,42,39)(37,44,41,43);(5,6,8,9)(7,10,11,12)(13,15)(14,16)(17,23)(18,22)(19,24)(20,21)(25,27)(26,28)(29,32,34,31)(30,36,38,35)(33,40,42,39)(37,44,41,43);(1,2)(3,4)(5,7,8,11)(6,10,9,12)(13,16)(14,15)(17,24)(18,21)(19,23)(20,22)(25,28)(26,27)(29,30)(31,36)(32,35)(33,41)(34,38)(37,42)(39,43)(40,44);(13,17,18,26)(14,19,20,25)(15,23,22,28)(16,24,21,27)(29,33)(30,37)(31,39)(32,40)(34,42)(35,43)(36,44)(38,41)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=gh,d^2=fgh,e^2=fg,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, 2983005);