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 | |
6126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(11,12)(14,16)(17,18)(19,20)(21,22)(23,24);(2,3)(6,7)(9,10)(11,16)(12,14)(13,15)(18,22)(20,23);(1,2,5,3)(4,6,8,7)(9,11,13,12)(10,14,15,16);(9,10)(11,16)(12,14)(13,15)(17,19)(18,23)(20,22)(21,24);(1,3,5,2)(4,6,8,7)(9,10)(11,14)(12,16)(13,15)(17,20,21,23)(18,19,22,24);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,14,15,16)(17,19)(18,20)(21,24)(22,23)Polycyclic presentation:
2^9|c^2=i,e^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72262899, #72262930, #72262946, … |
| maximal normal subgroup of | #72262899, #72262930, #72262946, … |
| maximal quotient of | #72278287, #72278296, #72278327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9756979);