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 | |
9635 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)(13,14)(15,16)(17,18)(19,20);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,15,19,17)(14,16,20,18);(5,7)(6,8)(9,11)(10,12)(13,16,19,18)(14,17,20,15);(13,15,19,17)(14,16,20,18);(5,7)(6,10)(8,12)(9,11)(13,17,19,15)(14,16,20,18);(5,8,9,10)(6,11,12,7)(13,18,19,16)(14,17,20,15)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=h,e^2=h,f^2=i,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 | #72233750, #72233755, #72233766, … |
| maximal normal subgroup of | #72233750, #72233755, #72233766, … |
| maximal quotient of | #72261044, #72261093, #72261133, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7786122);