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 | |
8141 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)(15,19)(16,20)(17,21)(18,22)(23,27)(24,26)(25,28);(1,2,3,4)(5,6)(7,10)(8,11)(9,12)(13,15)(14,19)(16,23)(17,22)(18,21)(20,27)(24,25)(26,28);(5,7)(6,9)(8,12)(10,11)(13,14,18,22)(15,17,21,19)(16,24,26,20)(23,27,28,25);(13,16,18,26)(14,20,22,24)(15,23,21,28)(17,25,19,27);(5,7)(6,10)(8,12)(9,11)(13,15)(14,17)(16,23)(18,21)(19,22)(20,25)(24,27)(26,28);(5,6)(7,9)(8,11)(10,12)(13,17)(14,21)(15,22)(16,25)(18,19)(20,28)(23,24)(26,27)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^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 | #72283867, #72284321, #72284836, … |
| maximal normal subgroup of | #72283867, #72284321, #72284836, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7757319);