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 | |
5537 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)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(18,22)(23,24);(2,7)(3,5)(17,18)(19,23)(20,24)(21,22);(1,3,6,5)(2,4,7,8)(10,14)(11,15)(17,19,21,20)(18,23,22,24);(1,4)(2,5)(3,7)(6,8)(9,11,13,15)(10,12,14,16);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,14)(13,16)(17,20,21,19)(18,24,22,23);(1,5,6,3)(2,8,7,4)Polycyclic presentation:
2^9|c^2=g,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72297317 |
| maximal normal subgroup of | #72297317 |
| maximal quotient of | #72297472, #72297522, #72297812 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7895177);