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:
(5,6,9,7)(8,11,12,10)(14,18)(16,23)(19,24)(22,27)(29,30)(31,34)(32,35)(33,38)(36,42)(37,40)(39,44)(41,43);(1,2,3,4)(5,7,9,6)(8,11,12,10)(14,18)(15,21)(20,25)(22,27)(30,31)(35,37)(36,38)(43,44);(1,2)(3,4)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28)(29,31,34,30)(32,37,40,35)(33,36,42,38)(39,43,41,44);(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,32,34,40)(30,35,31,37)(33,41,42,39)(36,44,38,43);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,33,34,42)(30,36,31,38)(32,39,40,41)(35,43,37,44)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=g,d^2=fi,e^2=f,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, 3421697);