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 | |
8098 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)(8,12)(10,11)(14,17)(15,22)(18,21)(24,26);(1,2)(3,4)(5,6,9,7)(8,10,12,11)(13,14,19,17)(15,23,22,20)(16,21,27,18)(24,28,26,25);(6,7)(10,11)(13,15,19,22)(14,20,17,23)(16,26,27,24)(18,25,21,28);(13,16,19,27)(14,21,17,18)(15,24,22,26)(20,25,23,28);(5,7,9,6)(8,11,12,10)(13,17,19,14)(15,20,22,23)(16,18,27,21)(24,25,26,28);(5,8)(6,10)(7,11)(9,12)(13,18)(14,16)(15,25)(17,27)(19,21)(20,26)(22,28)(23,24)Polycyclic presentation:
2^9|b^2=i,c^2=h,d^2=h,e^2=i,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 | #72304538, #72304570, #72304590, … |
| maximal normal subgroup of | #72304538, #72304570, #72304590, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7618408);