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 | |
8932 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(13,14)(15,16)(17,21)(18,19)(20,23)(22,24)(25,26)(27,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,19)(15,16,20,23)(17,25,26,21)(22,28,27,24);(6,9)(10,11)(14,19)(16,23)(21,25)(24,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,16,19,23)(17,22,26,27)(21,24,25,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,19)(17,24)(18,23)(21,27)(22,25)(26,28);(5,7)(6,10)(8,12)(9,11)(13,17)(14,21)(15,22)(16,24)(18,26)(19,25)(20,27)(23,28)Polycyclic presentation:
2^9|b^2=h,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 | #72304546, #72304580, #72304623, … |
| maximal normal subgroup of | #72304546, #72304580, #72304623, … |
| maximal quotient of | #72333726 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7640560);