Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3552 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2925 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14,19,23)(15,24,27,34)(16,21,29,37)(17,22,18,31)(20,32,35,26)(25,39,42,44)(28,36,30,38)(33,40,43,41);(5,6,10,9)(7,12,8,11)(13,15)(14,20)(16,25)(17,26)(18,32)(19,27)(21,33)(22,34)(23,35)(24,31)(28,40)(29,42)(30,41)(36,39)(37,43)(38,44);(2,4)(5,7,10,8)(6,11,9,12)(13,16,19,29)(14,21,23,37)(15,25,27,42)(17,30,18,28)(20,33,35,43)(22,38,31,36)(24,39,34,44)(26,41,32,40);(5,8,10,7)(6,11,9,12)(13,17,19,18)(14,22,23,31)(15,26,27,32)(16,28,29,30)(20,34,35,24)(21,36,37,38)(25,40,42,41)(33,39,43,44);(5,9,10,6)(7,12,8,11)(14,23)(20,35)(21,37)(22,31)(24,34)(33,43)(36,38)(39,44)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=hi,d^2=hi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,c^b=ch,f^b=fi,d^c=di,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479521);