Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34633 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(6,10)(11,12)(13,14,17,22,18,21,28,34)(15,19,25,31,24,30,37,41)(16,20,27,32,26,33,38,42)(23,29,36,40,35,39,43,44);(5,6,9,10)(7,11,8,12)(13,15,17,25,18,24,28,37)(14,19,22,31,21,30,34,41)(16,23,27,36,26,35,38,43)(20,29,32,40,33,39,42,44);(2,4)(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,23)(17,27)(18,26)(19,29)(21,33)(22,32)(24,35)(25,36)(28,38)(30,39)(31,40)(34,42)(37,43)(41,44);(5,8,9,7)(6,11,10,12)(14,21)(16,26)(19,30)(22,34)(23,35)(27,38)(31,41)(36,43);(15,24)(19,30)(23,35)(25,37)(29,39)(31,41)(36,43)(40,44)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=f,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289344);