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 |
| prime radical | C2×#3484 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9126 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)(6,10)(7,11)(13,14)(15,18)(16,23)(17,19)(20,24)(21,26)(22,28)(25,27);(1,2)(3,4)(5,6,9,10)(7,8,11,12)(13,15,17,21)(14,18,19,26)(16,22,24,27)(20,25,23,28);(5,7)(6,8)(9,11)(10,12)(14,19)(15,21)(20,23)(22,27);(5,8)(6,11)(7,10)(9,12)(13,14)(15,18)(16,20)(17,19)(21,26)(22,25)(23,24)(27,28);(13,15)(14,18)(16,22)(17,21)(19,26)(20,25)(23,28)(24,27);(13,16)(14,20)(15,22)(17,24)(18,25)(19,23)(21,27)(26,28)Polycyclic presentation:
2^9|b^2=h,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei110 conjugacy classes — view the full character table.
| maximal subgroup of | #72267576, #72267582, #72267587, … |
| maximal normal subgroup of | #72267576, #72267582, #72267587, … |
| maximal quotient of | #72331272, #72332117, #72332135, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7581899);