Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2220 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(6,9)(7,11)(13,14,19,24)(15,20,27,35)(16,21,30,38)(17,22,18,23)(25,33,42,44)(26,34,31,39)(28,36,41,43)(29,37,32,40);(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,28)(17,26)(18,31)(19,27)(21,36)(22,34)(23,39)(24,35)(25,29)(30,41)(32,42)(33,37)(38,43)(40,44);(2,4)(13,16)(14,21)(15,25)(17,29)(18,32)(19,30)(20,33)(22,37)(23,40)(24,38)(26,41)(27,42)(28,31)(34,43)(35,44)(36,39);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,24,23)(15,26,27,31)(16,29,30,32)(20,34,35,39)(21,37,38,40)(25,41,42,28)(33,43,44,36);(6,9)(10,12)(15,26)(17,18)(20,34)(22,23)(25,41)(27,31)(28,42)(29,32)(33,43)(35,39)(36,44)(37,40)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6578906);