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 | |
6224 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,4)(6,8)(17,18,22,23)(19,24,21,20);(9,10,13,15)(11,12,16,14)(18,23)(19,21);(1,2,5,4)(3,6,7,8)(9,11,13,16)(10,14,15,12)(18,23)(20,24);(3,7)(6,8)(9,11,13,16)(10,14,15,12)(17,19,22,21)(18,20,23,24);(1,3,5,7)(2,6,4,8)(9,11)(10,12)(13,16)(14,15)(17,20,22,24)(18,19,23,21);(1,4,5,2)(3,8,7,6)(9,12)(10,11)(13,14)(15,16)(17,21,22,19)(18,24,23,20)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=ghi,d^2=gi,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72327258 |
| maximal normal subgroup of | #72327258 |
| maximal quotient of | #72327437, #72327791, #72328247 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10275237);