Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,12)(8,10)(9,11)(13,14)(15,24)(16,22)(17,23)(18,31)(19,21)(20,26)(25,39)(27,35)(28,37)(29,36)(30,38)(32,33)(34,41)(40,43)(42,44);(1,2)(3,4)(13,15)(14,20)(16,25)(17,26)(18,32)(19,27)(21,33)(22,34)(23,35)(24,31)(28,40)(29,42)(30,41)(36,43)(37,39)(38,44);(2,4)(5,7,9,8)(6,10,11,12)(14,21)(20,33)(22,36)(23,31)(24,35)(34,43)(37,38)(39,44);(2,4)(5,8,9,7)(6,10,11,12)(13,16,19,29)(14,22,21,36)(15,25,27,42)(17,30,18,28)(20,34,33,43)(23,38,31,37)(24,39,35,44)(26,41,32,40);(13,17,19,18)(14,23,21,31)(15,26,27,32)(16,28,29,30)(20,35,33,24)(22,37,36,38)(25,40,42,41)(34,39,43,44)Polycyclic presentation:
2^9|c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6358913);