Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1720 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)(7,11)(9,12)(13,14,17,20)(15,21,22,18)(16,24,25,19)(23,26,28,27)(29,30)(31,32)(33,37)(34,35)(36,38)(39,40)(41,43)(42,44);(5,6,8,10)(7,9,11,12)(15,22)(16,25)(18,21)(19,24)(30,35)(31,38)(33,41)(40,42);(5,6)(7,9)(8,10)(11,12)(13,15,17,22)(14,18,20,21)(16,23,25,28)(19,26,24,27)(29,31)(30,36)(32,35)(33,39)(34,38)(37,42)(40,43)(41,44);(1,2)(3,4)(13,14)(15,21)(16,19)(17,20)(18,22)(23,27)(24,25)(26,28)(29,32,34,36)(30,31,35,38)(33,42,41,40)(37,44,43,39);(5,7)(6,9)(8,11)(10,12)(13,16,17,25)(14,19,20,24)(15,23,22,28)(18,26,21,27)(29,33,34,41)(30,37,35,43)(31,39,38,44)(32,40,36,42)Polycyclic presentation:
2^9|a^2=fg,b^2=gh,c^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5515728);