Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2200 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)(5,6,9,12)(7,8,11,10)(13,14)(15,20)(16,28)(17,32)(18,22)(19,23)(21,31)(24,41)(25,40)(26,35)(27,36)(29,44)(30,38)(33,39)(34,42)(37,43);(5,7,9,11)(6,10,12,8)(13,15,18,26)(14,20,22,35)(16,29,31,43)(17,27,19,25)(21,37,28,44)(23,40,32,36)(24,33,42,30)(34,38,41,39);(2,4)(5,8,9,10)(6,11,12,7)(13,16)(14,21)(15,24)(17,30)(18,31)(19,33)(20,34)(22,28)(23,38)(25,29)(26,42)(27,43)(32,39)(35,41)(36,44)(37,40);(2,4)(6,12)(7,11)(14,22)(20,35)(21,28)(23,32)(34,41)(36,40)(37,44)(38,39);(13,17,18,19)(14,23,22,32)(15,25,26,27)(16,30,31,33)(20,36,35,40)(21,38,28,39)(24,29,42,43)(34,44,41,37)Polycyclic presentation:
2^9|a^2=fgi,b^2=fi,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410960);