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 | |
8576 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)(5,6,8,10)(7,9,12,11)(13,14,19,21)(15,18,23,20)(16,17,22,28)(24,26,27,25);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,24)(17,25)(18,21)(19,23)(22,27)(26,28);(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(14,21)(15,23)(16,22)(25,26);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,22)(14,17,21,28)(15,24,23,27)(18,26,20,25);(13,17)(14,22)(15,25)(16,21)(18,24)(19,28)(20,27)(23,26);(13,18)(14,23)(15,21)(16,26)(17,27)(19,20)(22,25)(24,28)Polycyclic presentation:
2^9|a^2=g,c^2=gh,d^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72238149, #72238150, #72238250, … |
| maximal normal subgroup of | #72238149, #72238150, #72238250, … |
| maximal quotient of | #72297202, #72297624, #72297837 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7594268);