Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37436 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2030 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(6,9)(7,11)(13,14,17,22,18,21,28,34)(15,19,24,31,25,30,37,41)(16,20,27,33,26,32,38,42)(23,29,36,40,35,39,43,44);(5,6)(7,10)(8,9)(11,12)(13,15,17,24,18,25,28,37)(14,19,22,31,21,30,34,41)(16,23,27,36,26,35,38,43)(20,29,33,40,32,39,42,44);(1,2,3,4)(6,9)(10,12)(13,16)(14,20)(15,23)(17,27)(18,26)(19,29)(21,32)(22,33)(24,36)(25,35)(28,38)(30,39)(31,40)(34,42)(37,43)(41,44);(5,7,8,11)(6,10,9,12)(16,26)(20,32)(23,35)(27,38)(29,39)(33,42)(36,43)(40,44);(14,21)(19,30)(20,32)(22,34)(29,39)(31,41)(33,42)(40,44)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=cf,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6283432);