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 | |
7538 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(13,14)(15,20)(16,18)(17,27)(19,23)(21,26)(22,24)(25,28);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,15,19,25)(14,20,23,28)(16,24,26,17)(18,22,21,27);(5,7,9,10)(6,8,11,12)(13,16,19,26)(14,21,23,18)(15,17,25,24)(20,22,28,27);(13,16)(14,18)(15,24)(17,25)(19,26)(20,22)(21,23)(27,28);(5,6)(7,8)(9,11)(10,12)(13,17,19,24)(14,22,23,27)(15,16,25,26)(18,20,21,28);(5,8)(6,10)(7,11)(9,12)(13,18)(14,16)(15,22)(17,28)(19,21)(20,24)(23,26)(25,27)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=i,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72280398 |
| maximal normal subgroup of | #72280398 |
| maximal quotient of | #72280809, #72280918, #72281051, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7767126);