Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #47826 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, prime radical, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1694 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6,9,8)(7,10,11,12)(13,14,17,18)(15,21,22,19)(16,24,25,20)(23,26,28,27)(29,30,34,37)(31,38,32,35)(33,36,41,44)(39,43,40,42);(7,11)(10,12)(14,18)(15,22)(16,25)(26,27)(29,31,34,32)(30,35,37,38)(33,40,41,39)(36,43,44,42);(5,7)(6,10)(8,12)(9,11)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28)(29,32,34,31)(30,35,37,38)(33,40,41,39)(36,42,44,43);(1,2,4,3)(13,15)(14,19)(16,23)(17,22)(18,21)(20,26)(24,27)(25,28)(29,30)(31,35)(32,38)(33,36)(34,37)(39,42)(40,43)(41,44);(1,3,4,2)(5,8,9,6)(7,12,11,10)(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)(29,33,34,41)(30,36,37,44)(31,39,32,40)(35,42,38,43)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fg,d^2=fghi,e^2=i,b^a=bf,c^a=cg,d^a=df,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 478560);