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 | |
7342 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)(7,11)(8,9)(10,12)(13,14,17,21)(15,19,23,18)(16,20,25,28)(22,27,24,26);(1,2,3,4)(13,15)(14,18)(16,22)(17,23)(19,21)(20,26)(24,25)(27,28);(6,9)(10,11)(13,15,17,23)(14,18,21,19)(16,24,25,22)(20,27,28,26);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,27)(24,25)(26,28);(5,7)(6,10)(8,12)(9,11)(13,14)(15,18)(16,20)(17,21)(19,23)(22,26)(24,27)(25,28);(13,16,17,25)(14,20,21,28)(15,22,23,24)(18,26,19,27)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72312822, #72312831, #72312837, … |
| maximal normal subgroup of | #72312822, #72312831, #72312837, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7689926);