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 | |
8178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(7,10)(8,12)(13,14,18,16)(15,20,17,19);(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,15,18,17)(14,19,16,20);(5,7)(6,8)(9,10)(11,12)(14,16)(19,20);(5,6)(7,12)(8,10)(9,11)(13,16,18,14)(15,20,17,19);(5,8,9,12)(6,10,11,7)(13,17,18,15)(14,20,16,19);(5,6)(7,8)(9,11)(10,12)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,d^2=hi,e^2=h,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 | #72232192, #72234914, #72234927, … |
| maximal normal subgroup of | #72232192, #72234914, #72234927, … |
| maximal quotient of | #72232220, #72232548, #72232979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7784153);