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 |
| prime radical | #37679 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 44 points:
(1,2)(3,4)(6,10)(11,12)(17,30)(21,34)(25,39)(28,41)(32,36)(33,40)(37,44)(42,43);(5,6)(7,11)(8,12)(9,10)(13,14,18,22)(15,23,26,19)(16,20,29,27)(17,21,30,34)(24,35,38,31)(25,36,39,32)(28,33,41,40)(37,43,44,42);(2,4)(5,7,9,8)(6,11,10,12)(13,14)(15,19)(16,27)(17,21)(18,22)(20,29)(23,26)(24,35)(25,32)(28,40)(30,34)(31,38)(33,41)(36,39)(37,43)(42,44);(1,2)(3,4)(13,15,18,26)(14,19,22,23)(16,24,29,38)(17,25,30,39)(20,31,27,35)(21,32,34,36)(28,37,41,44)(33,42,40,43);(5,8,9,7)(6,12,10,11)(13,16)(14,20)(15,24)(17,28)(18,29)(19,31)(21,33)(22,27)(23,35)(25,37)(26,38)(30,41)(32,42)(34,40)(36,43)(39,44);(13,17)(14,21)(15,25)(16,28)(18,30)(19,32)(20,33)(22,34)(23,36)(24,37)(26,39)(27,40)(29,41)(31,42)(35,43)(38,44)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=i,e^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7580736);