Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 | |
1816 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)(15,25)(16,30)(17,22)(18,23)(19,24)(20,27)(21,32)(26,34)(28,35)(29,36)(31,38)(33,37)(39,40)(41,44)(42,43);(5,6)(7,10)(8,9)(11,12)(13,15,17,29,19,28,18,27)(14,20,23,35,24,36,22,25)(16,26,32,40,33,41,31,42)(21,34,30,43,38,44,37,39);(2,4)(6,9)(10,12)(13,16,17,32,19,33,18,31)(14,21,23,30,24,38,22,37)(15,26,29,40,28,41,27,42)(20,34,35,43,36,44,25,39);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,29)(16,31,33,32)(20,35,36,25)(21,30,38,37)(26,40,41,42)(34,39,44,43);(7,11)(10,12)(15,28)(20,36)(25,35)(26,41)(27,29)(34,44)(39,43)(40,42)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fi,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6460011);