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 | |
8222 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(14,18)(15,21)(20,24)(23,27);(1,2)(3,4)(13,14)(15,19)(16,24)(17,18)(20,25)(21,22)(23,28)(26,27);(5,6,8,10)(7,9,12,11)(13,14,17,18)(15,22,21,19)(16,20,25,24)(23,28,27,26);(5,6)(7,9)(8,10)(11,12)(13,15,17,21)(14,19,18,22)(16,23,25,27)(20,26,24,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,24)(17,18)(20,25)(21,22)(23,28)(26,27);(13,16)(14,20)(15,23)(17,25)(18,24)(19,26)(21,27)(22,28)Polycyclic presentation:
2^9|a^2=gi,c^2=h,d^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72284682, #72285084, #72285178, … |
| maximal normal subgroup of | #72284682, #72285084, #72285178, … |
| maximal quotient of | #72285096 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7887046);