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 | |
6291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,8)(3,7,6,4)(18,22)(23,24);(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22);(1,3)(2,4)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,20)(18,23,22,24);(1,4)(2,6)(3,8)(5,7)(9,10)(11,14)(12,15)(13,16);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,15)(14,16)(17,20,21,19)(18,24,22,23);(1,3)(2,7)(4,8)(5,6)(9,11)(10,14)(12,16)(13,15)Polycyclic presentation:
2^9|a^2=h,c^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72232195, #72232576, #72234894, … |
| maximal normal subgroup of | #72232195, #72232576, #72234894, … |
| maximal quotient of | #72232273, #72232489, #72232606, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7566526);