Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500689 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,3,4)(5,6,9,12)(7,8,11,10)(13,14)(15,19)(16,27)(17,20)(18,22)(21,28)(23,36)(24,38)(25,33)(26,31)(29,41)(30,34)(32,39)(35,42)(37,43)(40,44);(5,7,9,11)(6,10,12,8)(13,15,17,25)(14,19,20,33)(16,24,28,39)(18,23,30,26)(21,32,27,38)(22,36,34,31)(29,37,42,40)(35,44,41,43);(2,4)(5,8,9,10)(6,11,12,7)(14,20)(15,23)(18,30)(19,31)(21,27)(24,37)(25,26)(29,42)(32,43)(33,36)(38,44)(39,40);(2,4)(6,12)(7,11)(13,16)(14,21)(15,24)(17,28)(18,29)(19,32)(20,27)(22,35)(23,37)(25,39)(26,40)(30,42)(31,43)(33,38)(34,41)(36,44);(15,25)(16,28)(19,33)(21,27)(23,26)(29,42)(31,36)(35,41)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410963);