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 | |
8299 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)(14,19)(15,21)(20,27)(23,24);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,22)(16,20)(17,21)(18,19)(23,28)(24,25)(26,27);(5,7)(6,9)(8,12)(10,11)(13,15,18,21)(14,17,19,22)(16,24,26,23)(20,28,27,25);(13,16,18,26)(14,20,19,27)(15,23,21,24)(17,25,22,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,17)(16,20)(18,19)(21,22)(23,25)(24,28)(26,27);(5,6)(7,9)(8,11)(10,12)(13,17)(14,21)(15,19)(16,25)(18,22)(20,24)(23,27)(26,28)Polycyclic presentation:
2^9|c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72279324, C2×#7850121, #72283433, … |
| maximal normal subgroup of | #72279324, C2×#7850121, #72283433, … |
| maximal quotient of | C2×#7850121, #72280628, #72281164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7757317);