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 | |
1838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(7,11)(10,12)(13,14)(15,20)(16,24)(17,22)(18,19)(21,27)(23,26)(25,28)(29,30,34,37)(31,35,32,38)(33,41,42,36)(39,44,40,43);(6,9)(10,12)(14,19)(17,20)(21,24)(26,28)(29,31,34,32)(30,35,37,38)(33,39,42,40)(36,43,41,44);(5,6)(7,10)(8,9)(11,12)(13,15,18,22)(14,20,19,17)(16,25,27,23)(21,26,24,28)(29,32,34,31)(30,35,37,38)(33,40,42,39)(36,43,41,44);(1,2,3,4)(5,7)(6,10)(8,11)(9,12)(13,16,18,27)(14,21,19,24)(15,23,22,25)(17,26,20,28)(29,30)(31,38)(32,35)(33,36)(34,37)(39,44)(40,43)(41,42);(13,17,18,20)(14,22,19,15)(16,26,27,28)(21,25,24,23)(29,33)(30,36)(31,39)(32,40)(34,42)(35,43)(37,41)(38,44)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=fg,d^2=ghi,e^2=f,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, 1488579);