Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7786 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(6,10)(8,12)(13,14,18,19)(15,17,21,25)(16,20,23,26)(22,24,27,28);(1,2)(3,4)(6,10)(7,11)(16,23)(20,26)(22,27)(24,28);(5,6)(7,8)(9,10)(11,12)(14,19)(15,21)(20,26)(22,27);(6,10)(7,11)(13,15)(14,17)(16,22)(18,21)(19,25)(20,24)(23,27)(26,28);(5,7,9,11)(6,8,10,12)(13,16,18,23)(14,20,19,26)(15,22,21,27)(17,24,25,28);(5,8)(6,11)(7,10)(9,12)(13,17)(14,15)(16,24)(18,25)(19,21)(20,22)(23,28)(26,27)Polycyclic presentation:
2^9|a^2=ghi,e^2=h,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei92 conjugacy classes — view the full character table.
| maximal subgroup of | #72326930, #72330752, #72336756 |
| maximal normal subgroup of | #72326930, #72330752, #72336756 |
| maximal quotient of | #72326979, #72328319, #72329564, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7902642);