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 | #37659 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2210 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)(15,24,26,38)(16,21,29,35)(17,22,18,32)(20,27,34,28)(25,37,41,44)(30,36,31,42)(33,39,43,40);(5,7)(6,9)(8,12)(10,11)(13,15,18,28,19,26,17,27)(14,20,22,24,23,34,32,38)(16,25,31,40,29,41,30,39)(21,33,36,37,35,43,42,44);(2,4)(7,12)(9,10)(13,16)(14,21)(15,25)(17,30)(18,31)(19,29)(20,33)(22,36)(23,35)(24,37)(26,41)(27,39)(28,40)(32,42)(34,43)(38,44);(5,7)(6,10)(8,12)(9,11)(15,26)(16,29)(20,34)(21,35)(24,38)(27,28)(30,31)(36,42);(7,12)(9,10)(13,17,19,18)(14,22,23,32)(15,27,26,28)(16,30,29,31)(20,24,34,38)(21,36,35,42)(25,39,41,40)(33,37,43,44)Polycyclic presentation:
2^9|a^2=i,b^2=fh,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=eh80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6504265);