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 | |
1608 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)(13,14)(15,16)(17,25)(18,19)(20,26)(21,24)(22,27)(23,28)(29,30,34,32)(31,37,39,35)(33,36,42,40)(38,44,41,43);(6,10)(11,12)(14,19)(15,21)(17,26)(23,27)(29,31,34,39)(30,35,32,37)(33,41,42,38)(36,44,40,43);(5,6)(7,11)(8,12)(9,10)(13,15,18,21)(14,16,19,24)(17,22,26,28)(20,23,25,27)(29,31)(30,35)(32,37)(33,38)(34,39)(36,43)(40,44)(41,42);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,15)(17,27)(18,24)(19,21)(20,28)(22,25)(23,26)(29,32,34,30)(31,35,39,37)(33,36,42,40)(38,44,41,43);(5,8,9,7)(6,12,10,11)(13,17)(14,20)(15,22)(16,23)(18,26)(19,25)(21,28)(24,27)(29,33)(30,36)(31,38)(32,40)(34,42)(35,43)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=fgh,e^2=fh,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, 2989285);