Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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 | #35328 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2172 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)(7,11)(8,12)(9,10)(13,14)(15,20)(16,22)(17,31)(18,24)(19,23)(21,30)(25,35)(26,38)(27,37)(28,42)(29,40)(32,36)(33,41)(34,44)(39,43);(6,10)(11,12)(13,15)(14,20)(16,28)(17,26)(18,27)(19,32)(21,39)(22,42)(23,36)(24,37)(25,29)(30,43)(31,38)(33,44)(34,41)(35,40);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,25)(17,29)(18,30)(19,33)(20,34)(22,24)(23,40)(26,43)(27,44)(28,32)(31,41)(35,37)(36,39)(38,42);(5,8,9,7)(6,12,10,11)(13,16)(14,22)(15,25)(17,29)(18,30)(19,33)(20,35)(21,24)(23,41)(26,43)(27,44)(28,32)(31,40)(34,37)(36,42)(38,39);(13,17,18,19)(14,23,24,31)(15,26,27,32)(16,29,30,33)(20,36,37,38)(21,40,22,41)(25,43,44,28)(34,39,35,42)Polycyclic presentation:
2^9|c^2=fgi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,h^c=hi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351199);