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 | |
7546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(3,4)(6,8)(9,10)(11,16)(12,14)(13,15);(1,2)(3,8)(4,6)(5,7)(9,11,13,12)(10,14,15,16)(17,18,21,22)(19,23,24,20);(1,3,5,4)(2,6,7,8)(10,15)(14,16)(18,22)(20,23);(9,11,13,12)(10,14,15,16)(17,19)(18,20)(21,24)(22,23);(1,4,5,3)(2,6,7,8)(9,12,13,11)(10,16,15,14);(1,2)(3,8)(4,6)(5,7)(9,10)(11,14)(12,16)(13,15)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|b^2=hi,c^2=gh,d^2=g,e^2=h,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 | #72232519, #72233519, #72233536, … |
| maximal normal subgroup of | #72232519, #72233519, #72233536, … |
| maximal quotient of | #72232628, #72261261, #72327138, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8088789);