Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | |
1616 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14)(15,18)(16,21)(17,22)(19,24)(20,26)(23,27)(25,28)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(2,4)(14,15)(19,21)(20,22)(27,28)(29,30,33,36)(31,37,39,34)(32,35,41,40)(38,43,44,42);(1,2)(3,4)(5,6,9,8)(7,10,11,12)(13,15,18,14)(16,21,24,19)(17,20,26,22)(23,27,25,28)(29,30)(31,34)(32,35)(33,36)(37,39)(38,42)(40,41)(43,44);(5,7,9,11)(6,10,8,12)(13,16)(14,19)(15,21)(17,25)(18,24)(20,28)(22,27)(23,26)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(5,8,9,6)(7,12,11,10)(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=i,d^2=fi,e^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3700324);