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 | |
7255 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)(3,5)(4,8)(6,7)(10,13)(14,15);(1,3,6,8)(2,4,7,5)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24);(1,4,6,5)(2,3,7,8)(17,19)(18,21)(20,24)(22,23);(2,7)(3,8)(9,10)(11,15)(12,13)(14,16)(17,19)(18,22)(20,24)(21,23);(1,5,6,4)(2,8,7,3);(1,2)(3,4)(5,8)(6,7)(9,11)(10,14)(12,16)(13,15)(17,18)(19,21)(20,23)(22,24)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72233520, #72233527, #72233606, … |
| maximal normal subgroup of | #72233520, #72233527, #72233606, … |
| maximal quotient of | #72261326, #72327618, #72328617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8329406);