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 | |
8508 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)(13,14)(15,23)(16,17)(18,20)(19,22)(21,28)(24,25)(26,27);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12);(5,7)(6,8)(9,11)(10,12)(13,15,19,18)(14,20,22,23)(16,26,21,24)(17,25,28,27);(13,16,19,21)(14,17,22,28)(15,24,18,26)(20,27,23,25);(5,7)(6,10)(8,12)(9,11)(13,17)(14,21)(15,25)(16,22)(18,27)(19,28)(20,24)(23,26);(5,8,9,10)(6,11,12,7)(13,18,19,15)(14,20,22,23)(16,26,21,24)(17,27,28,25)Polycyclic presentation:
2^9|c^2=h,d^2=h,f^2=hi,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 | #72256671, #72256675, #72256680, … |
| maximal normal subgroup of | #72256671, #72256675, #72256680, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7786386);