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 | |
7702 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)(5,6,8,10)(7,9,11,12)(18,32)(22,36)(27,41)(29,35)(30,42)(34,38)(39,43)(40,44);(5,6)(7,9)(8,10)(11,12)(13,14,19,23)(15,24,28,20)(16,17,21,31)(18,22,32,36)(25,37,33,26)(27,38,41,34)(29,30,35,42)(39,44,43,40);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15,19,28)(14,20,23,24)(16,25,21,33)(17,26,31,37)(18,27,32,41)(22,34,36,38)(29,39,35,43)(30,40,42,44);(13,16)(14,17)(15,25)(18,29)(19,21)(20,26)(22,30)(23,31)(24,37)(27,39)(28,33)(32,35)(34,40)(36,42)(38,44)(41,43);(13,17)(14,21)(15,26)(16,23)(18,30)(19,31)(20,33)(22,35)(24,25)(27,40)(28,37)(29,36)(32,42)(34,43)(38,39)(41,44);(5,7,8,11)(6,9,10,12)(13,18)(14,22)(15,27)(16,29)(17,30)(19,32)(20,34)(21,35)(23,36)(24,38)(25,39)(26,40)(28,41)(31,42)(33,43)(37,44)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538012);