Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
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 | |
6706 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,16)(12,14)(13,15)(17,18)(19,24)(20,22)(21,23);(1,2,6,5)(3,8,4,7)(9,11,13,12)(10,14,15,16)(17,18)(19,24)(20,22)(21,23);(1,3,6,4)(2,7,5,8)(9,10)(11,16)(12,14)(13,15)(17,19,21,20)(18,22,23,24);(17,18)(19,24)(20,22)(21,23);(1,4,6,3)(2,7,5,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,22)(20,24)(21,23);(1,5,6,2)(3,8,4,7)(9,12,13,11)(10,14,15,16)(17,20,21,19)(18,22,23,24)Polycyclic presentation:
2^9|b^2=hi,c^2=gi,e^2=ghi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72233505, #72233523, #72233598, … |
| maximal normal subgroup of | #72233505, #72233523, #72233598, … |
| maximal quotient of | #72261034, #72303667, #72304099 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7591951);