Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
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 | |
8550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(8,12)(10,11)(14,17)(15,21)(20,25)(22,27);(1,2)(3,4)(6,7)(10,11)(16,24)(20,25)(22,27)(26,28);(5,6,9,7)(8,10,12,11)(13,14,18,17)(15,19,21,23)(16,20,24,25)(22,26,27,28);(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28);(5,7,9,6)(8,11,12,10)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(5,8)(6,10)(7,11)(9,12)(13,17,18,14)(15,23,21,19)(16,25,24,20)(22,28,27,26)Polycyclic presentation:
2^9|c^2=hi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72284505, #72286018, #72286224, … |
| maximal normal subgroup of | #72284505, #72286018, #72286224, … |
| maximal quotient of | #72305137 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549189);