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 | |
1610 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,12)(7,10,8,11)(14,18)(15,21)(16,23)(26,28)(29,30,34,35)(31,32,37,36)(33,41,44,38)(39,43,42,40);(1,2,3,4)(14,18)(15,21)(20,25)(22,27)(30,35)(31,37)(38,41)(39,42);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,37)(30,36,35,32)(33,42,44,39)(38,40,41,43);(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,36)(30,37,35,31)(33,43,44,40)(38,39,41,42);(5,8,9,7)(6,11,12,10)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33)(30,38)(31,39)(32,40)(34,44)(35,41)(36,43)(37,42)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,c^2=fi,d^2=f,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, 3604069);