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 | |
1628 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)(5,6,9,8)(7,10,11,12)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26)(29,30)(31,38)(32,36)(33,35)(34,41)(37,42)(39,43)(40,44);(7,11)(10,12)(14,18)(16,24)(19,21)(22,28)(29,31,35,32)(30,36,33,38)(34,40,42,39)(37,44,41,43);(5,7,9,11)(6,10,8,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27)(29,32,35,31)(30,36,33,38)(34,40,42,39)(37,43,41,44);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,14)(15,21)(16,25)(17,18)(19,23)(20,24)(22,26)(27,28)(29,33,35,30)(31,38,32,36)(34,37,42,41)(39,43,40,44);(5,8,9,6)(7,12,11,10)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)(29,34,35,42)(30,37,33,41)(31,39,32,40)(36,43,38,44)Polycyclic presentation:
2^9|a^2=gh,b^2=fg,c^2=fh,d^2=fgh,e^2=fgh,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, 2992743);