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 | |
6310 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(10,12)(14,16)(17,18)(19,24)(20,23)(21,22);(1,2)(3,7)(4,5)(6,8)(17,19,22,20)(18,23,21,24);(2,5)(6,7)(9,10,13,12)(11,14,15,16)(17,18,22,21)(19,23,20,24);(2,5)(6,7)(11,15)(14,16)(17,18,22,21)(19,24,20,23);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,12,16)(17,20,22,19)(18,23,21,24);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,16,15,14)(17,21,22,18)(19,23,20,24)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72233989, #72238553, #72238560, … |
| maximal normal subgroup of | #72233989, #72238553, #72238560, … |
| maximal quotient of | #72234008, #72234020, #72263396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10230924);