Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574015 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1762 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)(15,20)(16,27)(17,22)(18,32)(19,23)(21,29)(24,38)(25,34)(26,35)(28,37)(30,41)(31,36)(33,39)(40,43)(42,44);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,24)(17,31)(18,25)(19,26)(21,33)(22,36)(23,35)(27,38)(28,42)(29,39)(30,40)(32,34)(37,44)(41,43);(6,9)(7,11)(13,16,18,29,19,30,17,28)(14,21,22,27,23,37,32,41)(15,24,25,39,26,40,31,42)(20,33,36,38,35,44,34,43);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,32)(15,25,26,31)(16,28,30,29)(20,34,35,36)(21,27,37,41)(24,39,40,42)(33,43,44,38);(6,9)(7,11)(15,26)(20,35)(24,40)(25,31)(33,44)(34,36)(38,43)(39,42)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=g,d^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513636);