Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
7164 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,3)(6,7)(10,13)(14,16);(1,2,5,3)(4,6,8,7)(9,10,12,13)(11,14,15,16)(17,18,21,20)(19,23,24,22);(1,2,5,3)(4,7,8,6)(11,15)(14,16)(18,20)(22,23);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19)(18,22)(20,23)(21,24);(1,3,5,2)(4,7,8,6);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^9|b^2=gi,c^2=h,d^2=h,e^2=h,f^2=gi,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 | #72233555, #72233557, #72233607, … |
| maximal normal subgroup of | #72233555, #72233557, #72233607, … |
| maximal quotient of | #72278257, #72278267, #72278635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8103004);