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 | |
6276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10,14,16)(11,15,13,12)(17,18)(19,24)(20,22)(21,23);(1,2,5,8)(3,4,7,6)(9,11,14,13)(10,12,16,15)(17,19,21,22)(18,20,23,24);(1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,15)(14,16);(9,12)(10,13)(11,16)(14,15)(17,19,21,22)(18,20,23,24);(1,3)(2,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24);(1,4)(2,7)(3,8)(5,6)(9,13,14,11)(10,15,16,12)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=h,b^2=gi,d^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72256088, #72262986, #72262991, … |
| maximal normal subgroup of | #72256088, #72262986, #72262991, … |
| maximal quotient of | #72262657, #72326419, #72326830, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9583855);