Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C214 |
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 | |
11457 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(9,10,14,16)(11,12,17,15)(13,18,22,23)(19,20,24,21);(5,6)(7,8)(9,11,14,17)(10,15,16,12)(13,19,22,24)(18,21,23,20);(1,2,3,4)(10,16)(11,17)(18,23)(19,24);(1,2)(3,4)(9,10)(11,15)(12,17)(13,18)(14,16)(19,21)(20,24)(22,23);(9,12,14,15)(10,17,16,11)(13,21,22,20)(18,19,23,24);(9,13,14,22)(10,18,16,23)(11,19,17,24)(12,20,15,21)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72234479, #72234843 |
| maximal normal subgroup of | #72234479, #72234843 |
| maximal quotient of | #72281016, #72284598 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7605561);