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 | |
8324 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)(5,6)(7,8)(9,10)(11,12)(14,18)(15,21)(20,25)(23,27);(1,2)(3,4)(14,18)(16,24)(19,22)(23,27);(6,10)(7,11)(13,14)(15,22)(16,20)(17,18)(19,21)(23,28)(24,25)(26,27);(5,7,9,11)(6,8,10,12);(5,8)(6,11)(7,10)(9,12)(13,15,17,21)(14,19,18,22)(16,23,24,27)(20,26,25,28);(13,16,17,24)(14,20,18,25)(15,23,21,27)(19,26,22,28)Polycyclic presentation:
2^9|d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72257321, #72282025, #72282108, … |
| maximal normal subgroup of | #72257321, #72282025, #72282108, … |
| maximal quotient of | #72257907, #72311733 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7646511);