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 | |
7699 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2)(3,4)(6,9)(10,12)(13,14)(15,19)(16,23)(17,18)(20,24)(21,25)(22,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,14,17,18)(15,19,21,25)(16,20,24,23)(22,26,27,28);(2,4)(14,18)(15,21)(20,23)(22,27);(1,2)(3,4)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,28)(24,27);(5,7,8,11)(6,10,9,12)(13,16,17,24)(14,20,18,23)(15,22,21,27)(19,26,25,28)Polycyclic presentation:
2^9|b^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72327640 |
| maximal normal subgroup of | #72327640 |
| maximal quotient of | #72327859, #72328063, #72328238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7580319);