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 | #35328 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6982 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)(6,10)(7,12)(13,14,19,20)(15,21,24,34)(16,17,22,30)(18,32,33,23)(25,35,28,26)(27,41,38,36)(29,43,37,31)(39,40,42,44);(5,6)(7,11)(8,12)(9,10)(14,20)(15,24)(17,30)(23,32)(25,28)(27,38)(31,43)(39,42);(1,2,3,4)(5,7,9,12)(6,11,10,8)(13,15)(14,21)(16,28)(17,26)(18,27)(19,24)(20,34)(22,25)(23,36)(29,42)(30,35)(31,40)(32,41)(33,38)(37,39)(43,44);(5,8,9,11)(6,12,10,7)(13,16,19,22)(14,17,20,30)(15,25,24,28)(18,29,33,37)(21,35,34,26)(23,31,32,43)(27,39,38,42)(36,44,41,40);(13,17)(14,22)(15,26)(16,20)(18,31)(19,30)(21,25)(23,37)(24,35)(27,40)(28,34)(29,32)(33,43)(36,39)(38,44)(41,42);(13,18,19,33)(14,23,20,32)(15,27,24,38)(16,29,22,37)(17,31,30,43)(21,36,34,41)(25,39,28,42)(26,40,35,44)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=g,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7537859);