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 | |
8557 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)(13,14)(15,21)(16,20)(17,19)(18,23)(22,27)(24,26)(25,28);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,18)(16,22)(17,23)(19,21)(20,25)(24,28)(26,27);(5,7)(6,9)(8,12)(10,11)(14,19)(18,21)(20,26)(25,27);(13,14)(15,18)(16,20)(17,19)(21,23)(22,25)(24,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,22)(17,23)(19,21)(20,25)(24,28)(26,27);(5,6)(7,9)(8,11)(10,12)(13,16,17,24)(14,20,19,26)(15,22,23,28)(18,25,21,27)Polycyclic presentation:
2^9|b^2=gh,f^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 | #72238142, #72238218, #72238236, … |
| maximal normal subgroup of | #72238142, #72238218, #72238236, … |
| maximal quotient of | #72303430 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7758279);