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 | |
6053 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(10,13)(14,15)(17,18)(19,20)(21,22)(23,24);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(18,22)(19,23);(1,3)(2,4)(5,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,21,23)(18,20,22,24);(2,6)(4,7)(9,10)(11,15)(12,13)(14,16)(17,20)(18,23)(19,22)(21,24);(1,3)(2,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16);(1,4,5,7)(2,8,6,3)(9,11,12,16)(10,14,13,15)Polycyclic presentation:
2^9|c^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72283347, #72297480, #72300459, … |
| maximal normal subgroup of | #72283347, #72297480, #72300459, … |
| maximal quotient of | #72297567, #72297670, #72297827, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8606112);