Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| 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×#3562 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3070 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)(13,14)(15,20)(16,26)(17,22)(18,31)(19,23)(21,24)(25,34)(27,39)(28,33)(29,37)(30,41)(32,36)(35,38)(40,43)(42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,18)(14,20,23,31)(16,24,28,32)(17,25,29,30)(21,33,36,26)(22,34,37,41)(27,38,40,42)(35,43,44,39);(6,9)(7,11)(13,16,19,28)(14,21,23,36)(15,24,18,32)(17,27,29,40)(20,33,31,26)(22,35,37,44)(25,38,30,42)(34,43,41,39);(5,7,8,11)(6,10,9,12)(13,17,19,29)(14,22,23,37)(15,25,18,30)(16,27,28,40)(20,34,31,41)(21,35,36,44)(24,38,32,42)(26,39,33,43);(6,9)(10,12)(17,29)(22,37)(25,30)(27,40)(34,41)(35,44)(38,42)(39,43)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dh,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420534);