Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,10)(7,12)(13,14,17,18)(15,21,22,19)(16,20,24,26)(23,28,25,27)(29,30)(31,35)(32,34)(33,40)(36,39)(37,42)(38,44)(41,43);(5,6)(7,11)(8,12)(9,10)(14,18)(15,22)(16,24)(27,28)(29,31,34,39)(30,35,32,36)(33,41,42,38)(37,44,40,43);(5,7,9,12)(6,11,10,8)(13,15,17,22)(14,19,18,21)(16,25,24,23)(20,28,26,27)(29,31)(30,36)(32,35)(33,38)(34,39)(37,44)(40,43)(41,42);(1,2)(3,4)(5,8)(6,7)(9,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,22)(23,27)(24,26)(25,28)(29,32,34,30)(31,36,39,35)(33,40,42,37)(38,44,41,43);(5,7)(6,8)(9,12)(10,11)(13,16,17,24)(14,20,18,26)(15,23,22,25)(19,27,21,28)(29,33)(30,37)(31,38)(32,40)(34,42)(35,43)(36,44)(39,41)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=f,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5128778);