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 | |
6725 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)(13,14,18,19)(15,20,23,24)(16,27,29,21)(17,30,31,22)(25,37,35,32)(26,38,36,33)(28,34,41,40)(39,42,43,44);(1,2)(3,4)(14,19)(15,23)(21,27)(22,30)(25,35)(26,36)(34,40)(39,43);(6,9)(10,11)(13,14)(15,24)(16,27)(17,22)(18,19)(20,23)(21,29)(25,32)(26,38)(28,40)(30,31)(33,36)(34,41)(35,37)(39,42)(43,44);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,25)(17,26)(18,23)(19,24)(21,32)(22,33)(27,37)(28,39)(29,35)(30,38)(31,36)(34,42)(40,44)(41,43);(5,7,8,12)(6,10,9,11)(13,16,18,29)(14,21,19,27)(15,25,23,35)(17,28,31,41)(20,32,24,37)(22,34,30,40)(26,39,36,43)(33,42,38,44);(13,17,18,31)(14,22,19,30)(15,26,23,36)(16,28,29,41)(20,33,24,38)(21,34,27,40)(25,39,35,43)(32,42,37,44)Polycyclic presentation:
2^9|a^2=i,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, 7622709);