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 | |
8887 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,16)(17,19)(18,20);(1,2,3,4)(5,6,9,7)(8,11,12,10)(13,15)(14,16)(17,20)(18,19);(6,7)(10,11)(13,16,18,17)(14,15,20,19);(5,6,9,7)(8,10,12,11)(13,15)(14,17)(16,20)(18,19);(5,7,9,6)(8,10,12,11)(13,15)(14,16)(17,20)(18,19);(5,8,9,12)(6,10,7,11)(13,17,18,16)(14,19,20,15)Polycyclic presentation:
2^9|b^2=g,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72233538, #72233539, #72233584, … |
| maximal normal subgroup of | #72233538, #72233539, #72233584, … |
| maximal quotient of | #72261246, #72261631, #72261887 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7892481);