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 | |
8818 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)(7,11)(10,12)(13,14,18,19)(15,20,22,17)(16,24,26,21)(23,25,28,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12);(6,9)(10,12)(14,19)(15,22)(21,24)(23,28);(13,15)(14,20)(16,23)(17,19)(18,22)(21,27)(24,25)(26,28);(5,6)(7,10)(8,9)(11,12)(13,16,18,26)(14,21,19,24)(15,23,22,28)(17,25,20,27);(5,7)(6,10)(8,11)(9,12)(13,17,18,20)(14,15,19,22)(16,25,26,27)(21,23,24,28)Polycyclic presentation:
2^9|a^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304537, #72304569, #72304584, … |
| maximal normal subgroup of | #72304537, #72304569, #72304584, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7616241);