Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470395 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(7,11)(10,12)(13,14,19,24)(15,25,30,41)(16,21,31,38)(17,22,32,39)(18,33,26,23)(20,34,37,29)(27,40,42,36)(28,35,43,44);(5,6)(7,10)(8,9)(11,12)(13,15,26,34,19,30,18,29)(14,20,33,41,24,37,23,25)(16,27,32,43,31,42,17,28)(21,35,22,36,38,44,39,40);(2,4)(6,9)(10,12)(13,16)(14,21)(15,27)(17,18)(19,31)(20,35)(22,33)(23,39)(24,38)(25,40)(26,32)(28,29)(30,42)(34,43)(36,41)(37,44);(6,9)(10,12)(13,17)(14,22)(15,28)(16,26)(18,31)(19,32)(20,36)(21,23)(24,39)(25,35)(27,34)(29,42)(30,43)(33,38)(37,40)(41,44);(5,7,8,11)(6,10,9,12)(16,31)(17,32)(21,38)(22,39)(27,42)(28,43)(35,44)(36,40)Polycyclic presentation:
2^9|a^2=i,b^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398989);