Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2126 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)(5,6)(7,10)(8,9)(11,12)(13,14,19,25)(15,26,29,21)(16,31,17,22)(18,24,20,35)(23,33,40,34)(27,36,32,39)(28,42,30,38)(37,43,41,44);(6,9)(10,12)(13,15)(14,21)(16,32)(17,27)(18,28)(19,29)(20,30)(22,39)(23,37)(24,38)(25,26)(31,36)(33,44)(34,43)(35,42)(40,41);(1,2,3,4)(13,16,19,17)(14,22,25,31)(15,27,29,32)(18,34,20,33)(21,36,26,39)(23,24,40,35)(28,44,30,43)(37,42,41,38);(13,17,19,16)(14,23,25,40)(15,27,29,32)(18,34,20,33)(21,37,26,41)(22,24,31,35)(28,43,30,44)(36,42,39,38);(5,7,8,11)(6,10,9,12)(13,18,19,20)(14,24,25,35)(15,28,29,30)(16,33,17,34)(21,38,26,42)(22,23,31,40)(27,43,32,44)(36,41,39,37)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=i,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,c^b=ci,e^c=ei,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537449);