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 | |
6732 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 44 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14,19,24)(15,16,21,20)(17,29,32,22)(18,33,34,23)(25,35,30,27)(26,36,37,28)(31,38,43,41)(39,40,42,44);(1,2,3,4)(15,21)(16,20)(25,30)(26,37)(27,35)(28,36)(39,42)(40,44);(6,10)(7,11)(13,15)(14,20)(16,24)(17,30)(18,26)(19,21)(22,27)(23,36)(25,32)(28,33)(29,35)(31,42)(34,37)(38,40)(39,43)(41,44);(5,7,9,11)(6,8,10,12)(13,16,19,20)(14,21,24,15)(17,27,32,35)(18,28,34,36)(22,30,29,25)(23,37,33,26)(31,40,43,44)(38,42,41,39);(5,8)(6,11)(7,10)(9,12)(13,17,19,32)(14,22,24,29)(15,25,21,30)(16,27,20,35)(18,31,34,43)(23,38,33,41)(26,39,37,42)(28,40,36,44);(13,18,19,34)(14,23,24,33)(15,26,21,37)(16,28,20,36)(17,31,32,43)(22,38,29,41)(25,39,30,42)(27,40,35,44)Polycyclic presentation:
2^9|a^2=i,b^2=g,d^2=h,e^2=i,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, 7622877);