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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1626 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)(5,6,9,8)(7,10,11,12)(13,14,18,16)(15,21,22,19)(17,24,25,20)(23,26,28,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(7,11)(10,12)(14,16)(15,22)(17,25)(26,27)(29,30,33,36)(31,34,39,37)(32,40,41,35)(38,43,44,42);(5,7)(6,10)(8,12)(9,11)(13,15,18,22)(14,19,16,21)(17,23,25,28)(20,26,24,27)(29,30)(31,37)(32,35)(33,36)(34,39)(38,43)(40,41)(42,44);(1,2)(3,4)(5,8,9,6)(7,12,11,10)(13,16,18,14)(15,19,22,21)(17,24,25,20)(23,26,28,27)(29,31)(30,34)(32,38)(33,39)(35,42)(36,37)(40,43)(41,44);(13,17,18,25)(14,20,16,24)(15,23,22,28)(19,26,21,27)(29,32,33,41)(30,35,36,40)(31,38,39,44)(34,42,37,43)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=g,d^2=fgh,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, 1644806);