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 | |
1624 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6,9,7)(8,11,12,10)(14,18)(16,23)(19,24)(22,27)(29,30)(31,35)(32,36)(33,34)(37,39)(38,42)(40,41)(43,44);(1,2,3,4)(5,6,9,7)(8,10,12,11)(15,21)(19,24)(22,27)(26,28)(30,34)(32,40)(35,37)(38,44);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)(29,30)(31,37)(32,36)(33,34)(35,39)(38,43)(40,41)(42,44);(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28)(29,31)(30,35)(32,38)(33,39)(34,37)(36,42)(40,44)(41,43);(13,16,17,23)(14,20,18,25)(15,22,21,27)(19,26,24,28)(29,32,33,40)(30,36,34,41)(31,38,39,44)(35,42,37,43)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,c^2=i,d^2=i,e^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, 1732476);