Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574298 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #37682 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,24)(16,21)(17,22)(18,32)(19,23)(20,26)(25,38)(27,34)(28,36)(29,37)(30,42)(31,35)(33,39)(40,43)(41,44);(5,7)(6,9)(8,12)(10,11)(13,15,17,31,19,27,18,26)(14,20,32,34,23,35,22,24)(16,25,28,41,29,40,30,39)(21,33,42,43,37,44,36,38);(1,2,3,4)(7,12)(9,10)(13,16,19,29)(14,21,23,37)(15,25,27,40)(17,28,18,30)(20,33,35,44)(22,36,32,42)(24,38,34,43)(26,39,31,41);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,22,23,32)(15,26,27,31)(16,28,29,30)(20,34,35,24)(21,36,37,42)(25,39,40,41)(33,43,44,38);(7,12)(9,10)(15,27)(16,29)(20,35)(21,37)(24,34)(26,31)(28,30)(36,42)Polycyclic presentation:
2^9|b^2=fhi,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ehi,d^c=dh,e^c=ei,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6514093);