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 | |
7821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(6,10)(7,11)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(1,2)(3,4)(5,6)(7,12)(8,11)(9,10);(5,7,9,11)(6,8,10,12)(13,14,17,20)(15,18,23,21)(16,19,24,26)(22,25,28,27);(5,6)(7,12)(8,11)(9,10)(13,14)(15,18)(16,19)(17,20)(21,23)(22,25)(24,26)(27,28);(5,6)(7,8)(9,10)(11,12)(13,15)(14,18)(16,22)(17,23)(19,25)(20,21)(24,28)(26,27);(5,8)(6,11)(7,10)(9,12)(13,16,17,24)(14,19,20,26)(15,22,23,28)(18,25,21,27)Polycyclic presentation:
2^9|c^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72236851, #72236863, #72236865, … |
| maximal normal subgroup of | #72236851, #72236863, #72236865, … |
| maximal quotient of | #72300335, #72300434, #72331644, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7809479);