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 | |
6297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,4)(5,7)(6,8)(9,10,14,15)(11,12,16,13)(18,22)(20,24);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(18,22)(19,23);(9,12,14,13)(10,11,15,16)(17,18)(19,20)(21,22)(23,24);(2,7)(4,6)(10,15)(12,13)(18,22)(19,23);(1,4,5,6)(2,3,7,8)(9,13,14,12)(10,11,15,16)(17,19,21,23)(18,20,22,24);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,15)(14,16)(17,20)(18,19)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=hi,c^2=hi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232446, #72234803, #72262857, … |
| maximal normal subgroup of | #72232446, #72234803, #72262857, … |
| maximal quotient of | #72232527, #72234813, #72234826, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10065776);