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 | |
9874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(6,9)(7,11)(13,14,18,19)(15,21,22,16)(17,25,26,20)(23,24,28,27);(1,2)(3,4)(6,9)(10,12)(14,19)(16,21)(20,25)(24,27);(5,6)(7,10)(8,9)(11,12)(14,19)(15,22)(20,25)(23,28);(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,23)(18,22)(19,21)(20,24)(25,27)(26,28);(13,16)(14,15)(17,24)(18,21)(19,22)(20,23)(25,28)(26,27);(13,17)(14,20)(15,23)(16,24)(18,26)(19,25)(21,27)(22,28)Polycyclic presentation:
2^9|a^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^b=ehi,d^c=dhi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72267579, #72267588, #72284265, … |
| maximal normal subgroup of | #72267579, #72267588, #72284265, … |
| maximal quotient of | #72327794, #72328033, #72335280 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7549205);