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 | |
8644 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)(6,8)(7,11)(17,25)(20,27)(23,26)(24,28);(1,2)(3,4)(5,6,9,8)(7,12,11,10)(13,14)(15,22)(16,19)(17,20)(18,21)(23,28)(24,26)(25,27);(5,7,9,11)(6,10,8,12)(13,15,18,19)(14,16,21,22)(17,23,25,26)(20,24,27,28);(5,8,9,6)(7,12,11,10)(13,14)(15,16)(17,20)(18,21)(19,22)(23,24)(25,27)(26,28);(13,16)(14,19)(15,21)(17,24)(18,22)(20,26)(23,27)(25,28);(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,27)(22,28)Polycyclic presentation:
2^9|b^2=h,c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,e^d=ehi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72297209 |
| maximal normal subgroup of | #72297209 |
| maximal quotient of | #72297232, #72297507, #72297783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7552563);