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 |
| prime radical | #59041 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5316 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,6,4)(3,8,7,5)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,24)(21,23);(1,3)(2,5)(4,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,20)(18,22,23,24);(2,4)(3,7)(18,23)(22,24);(1,2,6,4)(3,5,7,8)(17,18)(19,22)(20,24)(21,23);(1,4,6,2)(3,8,7,5)(17,20,21,19)(18,24,23,22);(1,5,6,8)(2,7,4,3)(9,11,12,15)(10,13,14,16)Polycyclic presentation:
2^9|a^2=g,b^2=hi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72239243, #72239255 |
| maximal normal subgroup of | #72239243, #72239255 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7578517);