Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35634 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3325 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)(15,23)(16,21)(17,29)(18,20)(19,25)(22,31)(24,37)(26,36)(27,34)(28,41)(30,32)(33,39)(35,40)(38,44)(42,43);(5,6)(7,10)(8,9)(11,12)(13,15,18,26)(14,19,20,32)(16,24,27,38)(17,30,31,25)(21,33,34,43)(22,36,29,23)(28,42,40,39)(35,44,41,37);(2,4)(6,9)(7,11)(14,20)(19,32)(21,34)(22,29)(23,36)(33,43)(35,41)(37,44);(5,7,8,11)(6,10,9,12)(13,16,18,27)(14,21,20,34)(15,24,26,38)(17,28,31,40)(19,33,32,43)(22,35,29,41)(23,37,36,44)(25,39,30,42);(6,9)(10,12)(16,27)(21,34)(24,38)(28,40)(33,43)(35,41)(37,44)(39,42)Polycyclic presentation:
2^9|a^2=gi,b^2=i,d^2=hi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6424106);