Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(7,11)(10,12)(13,14)(15,18)(16,19)(17,20)(21,24)(22,26)(23,27)(25,28)(29,30,33,36)(31,37,38,34)(32,35,40,42)(39,43,44,41);(2,4)(5,6,9,8)(7,10,11,12)(14,15)(19,21)(20,22)(27,28)(31,38)(32,40)(34,37)(35,42);(1,2,3,4)(13,15,18,14)(16,19,24,21)(17,22,26,20)(23,27,25,28)(29,31)(30,34)(32,39)(33,38)(35,41)(36,37)(40,44)(42,43);(5,7)(6,10)(8,12)(9,11)(13,16,18,24)(14,19,15,21)(17,25,26,23)(20,28,22,27)(29,32,33,40)(30,35,36,42)(31,39,38,44)(34,41,37,43);(5,8,9,6)(7,12,11,10)(13,17,18,26)(14,20,15,22)(16,23,24,25)(19,27,21,28)(29,30)(31,34)(32,35)(33,36)(37,38)(39,41)(40,42)(43,44)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=fgh,d^2=g,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2698590);