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 | |
8598 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)(7,8)(9,10)(11,12)(13,14,17,18)(15,19,22,25)(16,20,24,23)(21,26,28,27);(1,2)(3,4)(13,14)(15,19)(16,23)(17,18)(20,24)(21,27)(22,25)(26,28);(6,10)(7,11)(14,18)(19,25)(20,23)(26,27);(5,7,9,11)(6,8,10,12)(13,15,17,22)(14,19,18,25)(16,21,24,28)(20,26,23,27);(5,8)(6,11)(7,10)(9,12)(13,14)(15,19)(16,20)(17,18)(21,26)(22,25)(23,24)(27,28);(13,16)(14,20)(15,21)(17,24)(18,23)(19,26)(22,28)(25,27)Polycyclic presentation:
2^9|a^2=i,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72238230, #72238234, #72238248, … |
| maximal normal subgroup of | #72238230, #72238234, #72238248, … |
| maximal quotient of | #72257802, #72257923, #72263515, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7641581);