Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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×#3521 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3150 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,25,28,37)(16,30,32,21)(17,24,19,22)(20,29,36,27)(26,41,42,38)(31,44,43,35)(33,39,34,40);(2,4)(13,15,18,28)(14,20,23,36)(16,31,32,43)(17,29,19,27)(21,38,30,41)(22,37,24,25)(26,33,42,34)(35,39,44,40);(6,9)(10,12)(13,16)(14,21)(15,26)(17,34)(18,32)(19,33)(20,35)(22,40)(23,30)(24,39)(25,38)(27,31)(28,42)(29,43)(36,44)(37,41);(15,27)(17,19)(20,25)(22,24)(26,31)(28,29)(33,34)(35,38)(36,37)(39,40)(41,44)(42,43);(5,7,8,11)(6,10,9,12)(16,32)(21,30)(26,42)(31,43)(33,34)(35,44)(38,41)(39,40)Polycyclic presentation:
2^9|a^2=i,b^2=i,e^2=gi,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361282);