Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3524 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2974 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)(13,14,17,23,19,22,18,24)(15,25,32,20,27,37,28,35)(16,29,33,21,31,38,30,36)(26,34,41,39,40,44,42,43);(2,4)(5,6,8,9)(7,10,11,12)(13,15,19,27)(14,20,22,35)(16,26,31,40)(17,28,18,32)(21,34,36,44)(23,25,24,37)(29,39,38,43)(30,41,33,42);(6,9)(7,11)(13,16)(14,21)(15,26)(17,30)(18,33)(19,31)(20,34)(22,36)(23,29)(24,38)(25,39)(27,40)(28,41)(32,42)(35,44)(37,43);(5,7)(6,10)(8,11)(9,12)(14,22)(15,27)(21,36)(23,24)(26,40)(28,32)(29,38)(41,42);(6,9)(7,11)(13,17,19,18)(14,23,22,24)(15,28,27,32)(16,30,31,33)(20,25,35,37)(21,29,36,38)(26,41,40,42)(34,39,44,43)Polycyclic presentation:
2^9|a^2=fi,b^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503697);