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 | |
7161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,15)(12,14)(13,16)(17,18,21,22)(19,23,24,20);(1,2,5,6)(3,7,8,4)(9,11,12,16)(10,13,14,15)(17,18,21,22)(19,23,24,20);(2,6)(4,7)(10,14)(13,15)(18,22)(20,23);(2,6)(3,8)(9,11,12,16)(10,13,14,15)(17,19,21,24)(18,23,22,20);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,20)(21,24)(22,23);(1,4)(2,3)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=gh,b^2=gi,d^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72232069, #72232280, #72234872, … |
| maximal normal subgroup of | #72232069, #72232280, #72234872, … |
| maximal quotient of | #72232095, #72232161, #72232317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8295003);