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 | |
6245 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)(3,7)(4,6)(5,8)(18,20)(22,24);(1,3)(2,5)(4,8)(6,7)(9,10,13,14)(11,15,16,12)(19,23)(22,24);(2,6)(5,7)(10,14)(12,15)(17,18,21,20)(19,22,23,24);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19,21,23)(18,22,20,24);(1,3)(2,5)(4,8)(6,7)(17,20,21,18)(19,24,23,22);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,15)(14,16)Polycyclic presentation:
2^9|b^2=i,c^2=gh,d^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232554, #72232841, #72262857, … |
| maximal normal subgroup of | #72232554, #72232841, #72262857, … |
| maximal quotient of | #72232705, #72232948, #72232962, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8089046);