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 | |
7019 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12);(1,2)(3,4)(5,7,11,9)(6,8,12,10)(13,14,17,20)(15,18,22,26)(16,19,24,28)(21,25,23,27);(5,8,11,10)(6,9,12,7)(13,14)(15,18)(16,19)(17,20)(21,25)(22,26)(23,27)(24,28);(5,7,11,9)(6,8,12,10);(5,9,11,7)(6,8,12,10)(13,15,17,22)(14,18,20,26)(16,23,24,21)(19,27,28,25);(5,10,11,8)(6,9,12,7)(13,16,17,24)(14,19,20,28)(15,21,22,23)(18,25,26,27)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,e^2=i,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 | #72284677, #72284790, #72315080, … |
| maximal normal subgroup of | #72284677, #72284790, #72315080, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7796167);