Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4379698 |
| 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 | |
1710 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)(6,9)(10,12)(13,14,18,16)(15,19,21,22)(17,24,25,20)(23,28,27,26)(30,34)(32,40)(35,37)(38,41);(15,21)(19,22)(23,27)(26,28)(29,30,33,34)(31,37,39,35)(32,36,40,42)(38,44,41,43);(1,2)(3,4)(13,15,18,21)(14,19,16,22)(17,23,25,27)(20,26,24,28)(29,31,33,39)(30,35,34,37)(32,41,40,38)(36,44,42,43);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,22,21,19)(17,24,25,20)(23,28,27,26)(29,30)(31,35)(32,36)(33,34)(37,39)(38,43)(40,42)(41,44);(5,7,8,11)(6,10,9,12)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)(29,32,33,40)(30,36,34,42)(31,38,39,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=fh,b^2=f,c^2=h,d^2=fh,e^2=gi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4315593);