Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1792 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)(9,12)(14,18)(16,23)(19,24)(22,27)(29,30,33,34)(31,37,38,35)(32,36,40,42)(39,44,41,43);(2,4)(5,6,8,10)(7,9,11,12)(14,18)(15,21)(20,25)(22,27)(30,34)(31,38)(32,40)(43,44);(1,2,3,4)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,31,33,38)(30,35,34,37)(32,41,40,39)(36,44,42,43);(5,7)(6,9)(8,11)(10,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28)(29,32)(30,36)(31,39)(33,40)(34,42)(35,43)(37,44)(38,41);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)(29,30)(31,35)(32,36)(33,34)(37,38)(39,43)(40,42)(41,44)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fgh,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1449169);