Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
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 | |
8637 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)(6,9)(7,11)(13,14,18,19)(15,23,24,20)(16,21,26,25)(17,22,27,28);(1,2)(3,4)(6,9)(10,12);(5,6)(7,10)(8,9)(11,12)(14,19)(15,24)(21,25)(22,28);(5,7,8,11)(6,10,9,12)(13,15,18,24)(14,20,19,23)(16,22,26,28)(17,25,27,21);(13,16)(14,21)(15,22)(17,20)(18,26)(19,25)(23,27)(24,28);(13,17)(14,22)(15,25)(16,23)(18,27)(19,28)(20,26)(21,24)Polycyclic presentation:
2^9|a^2=ghi,d^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72238142, #72238150, #72238222, … |
| maximal normal subgroup of | #72238142, #72238150, #72238222, … |
| maximal quotient of | #72297899, #72297979, #72298096, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7569262);