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 | |
7473 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)(10,11)(15,22)(20,26)(23,28)(24,27);(1,2)(3,4)(13,14)(15,20)(16,21)(17,25)(18,19)(22,26)(23,27)(24,28);(5,6,9,8)(7,11,12,10)(14,19)(16,25)(20,26)(23,28);(5,7,9,12)(6,10,8,11)(13,15)(14,20)(16,23)(17,24)(18,22)(19,26)(21,27)(25,28);(5,8,9,6)(7,11,12,10)(13,16,18,25)(14,21,19,17)(15,23,22,28)(20,27,26,24);(13,17,18,21)(14,16,19,25)(15,24,22,27)(20,23,26,28)Polycyclic presentation:
2^9|c^2=h,d^2=h,e^2=hi,f^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 | #72238150, #72238167, #72238189, … |
| maximal normal subgroup of | #72238150, #72238167, #72238189, … |
| maximal quotient of | #72297150, #72297472, #72297968 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7895403);