Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 |
| prime radical | #35634 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2437 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)(7,9)(8,10)(11,12)(13,14)(15,26)(16,21)(17,23)(18,24)(19,25)(20,30)(22,33)(27,43)(28,37)(29,42)(31,40)(32,39)(34,41)(35,38)(36,44);(5,7,9,6)(8,11,12,10)(13,15)(14,20)(16,31)(17,29)(18,30)(19,35)(21,37)(22,40)(23,38)(24,26)(25,42)(27,32)(28,33)(34,44)(36,39)(41,43);(2,4)(5,8,9,12)(6,10,7,11)(13,16)(14,21)(15,27)(17,32)(18,33)(19,34)(20,36)(22,24)(23,39)(25,41)(26,43)(28,29)(30,44)(31,35)(37,42)(38,40);(2,4)(6,7)(10,11)(13,16)(14,22)(15,28)(17,34)(18,33)(19,32)(20,37)(21,24)(23,39)(25,41)(26,40)(27,29)(30,31)(35,44)(36,42)(38,43);(13,17,18,19)(14,23,24,25)(15,29,30,35)(16,32,33,34)(20,38,26,42)(21,39,22,41)(27,28,44,31)(36,40,43,37)Polycyclic presentation:
2^9|b^2=fi,c^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7409049);