Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432838 |
| 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 | |
2086 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,20,26,34)(16,29,31,38)(17,23,18,22)(21,32,37,30)(25,39,40,44)(27,36,28,35)(33,42,43,41);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,35)(23,36)(24,34)(29,39)(30,41)(31,40)(32,42)(37,43)(38,44);(1,3,4,2)(7,11)(9,12)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,33)(22,29)(23,38)(24,37)(26,40)(27,41)(28,42)(34,43)(35,39)(36,44);(6,10)(9,12)(15,26)(20,34)(25,40)(27,28)(33,43)(35,36)(39,44)(41,42);(13,17,19,18)(14,22,24,23)(15,27,26,28)(16,30,31,32)(20,35,34,36)(21,29,37,38)(25,41,40,42)(33,39,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340399);