Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6574024 |
| 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 | #37659 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2240 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)(8,11)(9,12)(13,14,19,23)(15,24,26,40)(16,21,30,37)(17,22,18,33)(20,27,35,28)(25,39,29,43)(31,38,32,44)(34,41,36,42);(5,7)(6,9)(8,12)(10,11)(13,15,18,28,19,26,17,27)(14,20,22,24,23,35,33,40)(16,29,32,41,30,25,31,42)(21,36,38,43,37,34,44,39);(2,4)(7,12)(9,10)(13,16)(14,21)(15,25)(17,31)(18,32)(19,30)(20,34)(22,38)(23,37)(24,39)(26,29)(27,41)(28,42)(33,44)(35,36)(40,43);(5,7)(6,10)(8,12)(9,11)(15,26)(16,30)(20,35)(21,37)(24,40)(27,28)(31,32)(38,44);(7,12)(9,10)(13,17,19,18)(14,22,23,33)(15,27,26,28)(16,31,30,32)(20,24,35,40)(21,38,37,44)(25,41,29,42)(34,39,36,43)Polycyclic presentation:
2^9|a^2=i,b^2=fhi,e^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^d=ehi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504413);