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 | |
8610 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)(6,9)(7,11)(14,18)(15,21)(20,25)(22,27);(1,2)(3,4)(6,9)(10,12)(16,23)(20,25)(22,27)(26,28);(5,6,8,9)(7,10,11,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28);(5,7,8,11)(6,10,9,12)(13,15,17,21)(14,19,18,24)(16,22,23,27)(20,26,25,28);(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28);(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,25)(27,28)Polycyclic presentation:
2^9|c^2=i,d^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 | #72304778, #72304782, #72304797, … |
| maximal normal subgroup of | #72304778, #72304782, #72304797, … |
| maximal quotient of | #72305000, #72305042, #72305138 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549087);