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 | |
6667 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,8)(9,10)(11,15)(12,16)(13,14);(3,7)(6,8)(9,10,13,14)(11,15,16,12)(17,18)(19,20)(21,23)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,22)(20,23)(21,24);(9,12)(10,11)(13,15)(14,16)(17,20,21,22)(18,19,23,24);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,19)(18,20)(21,24)(22,23);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,22)(20,24)(21,23)Polycyclic presentation:
2^9|b^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72261846, #72261985, #72336552 |
| maximal normal subgroup of | #72261846, #72261985, #72336552 |
| maximal quotient of | #72336661 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9130299);