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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14,18,15)(16,21,24,19)(17,22,25,20)(23,27,26,28)(29,30)(31,34)(32,37)(33,36)(35,39)(38,41)(40,44)(42,43);(2,4)(6,9)(10,12)(14,15)(17,25)(19,21)(23,26)(30,31)(32,39)(33,41)(43,44);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15,18,14)(16,19,24,21)(17,22,25,20)(23,27,26,28)(29,31,34,30)(32,35,39,37)(33,38,41,36)(40,43,42,44);(5,7)(6,10)(8,11)(9,12)(13,16,18,24)(14,19,15,21)(17,26,25,23)(20,28,22,27)(29,32)(30,35)(31,37)(33,42)(34,39)(36,44)(38,43)(40,41);(13,17,18,25)(14,20,15,22)(16,23,24,26)(19,27,21,28)(29,33)(30,36)(31,38)(32,40)(34,41)(35,43)(37,44)(39,42)Polycyclic presentation:
2^9|a^2=fg,c^2=fhi,d^2=fg,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, 3014438);