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 | |
7223 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)(13,14,18,17)(15,21,23,19)(16,20,24,25)(22,27,28,26);(5,8,11,10)(6,9,12,7)(13,15,18,23)(14,19,17,21)(16,22,24,28)(20,26,25,27);(5,7,11,9)(6,8,12,10);(5,9,11,7)(6,8,12,10)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(5,10,11,8)(6,9,12,7)(13,17,18,14)(15,21,23,19)(16,20,24,25)(22,26,28,27)Polycyclic presentation:
2^9|b^2=i,c^2=i,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 | #72284652, #72284820, #72312421, … |
| maximal normal subgroup of | #72284652, #72284820, #72312421, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7790907);