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 | |
8340 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)(5,6,8,10)(7,9,11,12)(13,14,19,21)(15,24,25,20)(16,17,22,28)(18,27,26,23);(5,6)(7,9)(8,10)(11,12)(13,15,19,25)(14,20,21,24)(16,26,22,18)(17,27,28,23);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(14,21)(15,25)(16,22)(23,27);(13,16)(14,17)(15,26)(18,25)(19,22)(20,27)(21,28)(23,24);(13,17)(14,22)(15,23)(16,21)(18,24)(19,28)(20,26)(25,27);(5,7,8,11)(6,9,10,12)(13,18,19,26)(14,23,21,27)(15,16,25,22)(17,24,28,20)Polycyclic presentation:
2^9|a^2=g,b^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=di,e^d=ei128 conjugacy classes — view the full character table.
| maximal subgroup of | #72304537, #72304572, #72304616, … |
| maximal normal subgroup of | #72304537, #72304572, #72304616, … |
| maximal quotient of | #72316041 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7538240);