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 | |
9116 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(6,9)(10,11)(15,21)(16,23)(19,25)(20,24);(1,2)(3,4)(13,14)(15,19)(16,20)(17,18)(21,25)(22,26)(23,24)(27,28);(5,6)(7,11)(8,9)(10,12)(14,18)(19,25)(20,24)(26,28);(13,15)(14,19)(16,22)(17,21)(18,25)(20,26)(23,27)(24,28);(5,7,8,12)(6,10,9,11)(13,14)(15,19)(16,24)(17,18)(20,23)(21,25)(22,28)(26,27);(5,6)(7,11)(8,9)(10,12)(13,16)(14,20)(15,22)(17,23)(18,24)(19,26)(21,27)(25,28)Polycyclic presentation:
2^9|e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72267574, #72267579, #72267587 |
| maximal normal subgroup of | #72267574, #72267579, #72267587 |
| maximal quotient of | #72327758, #72327801, #72327909, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7571856);