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 | |
6213 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)(5,8)(6,7)(9,10,14,13)(11,16,15,12)(17,18,21,24)(19,22,23,20);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,16)(14,15)(17,19,21,23)(18,22,24,20);(1,4)(2,7)(3,8)(5,6)(17,19)(18,20)(21,23)(22,24);(2,8)(4,6)(9,12,14,16)(10,15,13,11)(17,19,21,23)(18,22,24,20);(1,4)(2,3)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,22)(20,24)(21,23);(1,3)(2,6)(4,8)(5,7)(9,13,14,10)(11,16,15,12)(17,20,21,22)(18,23,24,19)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=gh,f^2=gh,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 | #72336206, #72336905 |
| maximal normal subgroup of | #72336206, #72336905 |
| maximal quotient of | #72336343, #72336776, #72336979 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10004211);