Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7060 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 44 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,19,24)(15,25,26,20)(16,21,29,27)(17,30,33,22)(18,34,35,23)(28,36,40,39)(31,41,42,37)(32,38,44,43);(1,2,3,4)(15,26)(16,29)(20,25)(21,27)(28,40)(31,42)(36,39)(37,41);(6,9)(10,11)(13,15)(14,20)(16,30)(17,21)(18,28)(19,26)(22,29)(23,36)(24,25)(27,33)(31,43)(32,37)(34,39)(35,40)(38,42)(41,44);(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,22)(17,25)(18,31)(19,29)(20,33)(23,37)(24,27)(26,30)(28,38)(32,39)(34,41)(35,42)(36,44)(40,43);(5,7,8,12)(6,10,9,11)(13,17,19,33)(14,22,24,30)(15,27,26,21)(16,25,29,20)(18,32,35,44)(23,38,34,43)(28,41,40,37)(31,39,42,36);(13,18,19,35)(14,23,24,34)(15,28,26,40)(16,31,29,42)(17,32,33,44)(20,36,25,39)(21,37,27,41)(22,38,30,43)Polycyclic presentation:
2^9|a^2=i,b^2=g,e^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7622710);