Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438017 |
| 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 | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1997 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,22,18,20,30,34)(15,23,25,31,26,36,29,19)(16,21,28,35,27,33,40,43)(24,37,39,42,38,44,41,32);(1,2)(3,4)(13,15,18,26)(14,19,20,31)(16,24,27,38)(17,29,30,25)(21,32,33,42)(22,36,34,23)(28,41,40,39)(35,44,43,37);(2,4)(6,9)(10,12)(14,20)(19,31)(21,33)(22,34)(23,36)(32,42)(35,43)(37,44);(2,4)(13,16)(14,21)(15,24)(17,28)(18,27)(19,32)(20,33)(22,35)(23,37)(25,39)(26,38)(29,41)(30,40)(31,42)(34,43)(36,44);(5,7,8,11)(6,10,9,12)(16,27)(21,33)(24,38)(28,40)(32,42)(35,43)(37,44)(39,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,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, 6361706);