Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C211×C4 |
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 | |
6580 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(9,10)(11,16)(12,14)(13,15)(18,20)(22,24);(1,2)(3,4)(5,6)(7,8)(9,11,14,13)(10,15,12,16)(19,23)(22,24);(2,6)(4,7)(9,10,14,12)(11,15,13,16)(17,18,21,20)(19,22,23,24);(9,10,14,12)(11,16,13,15)(17,19)(18,22)(20,24)(21,23);(1,3)(2,7)(4,6)(5,8)(9,12,14,10)(11,15,13,16)(17,20,21,18)(19,24,23,22);(1,4,5,7)(2,3,6,8)(9,13,14,11)(10,15,12,16)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=g,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72233511, #72233542, #72233544, … |
| maximal normal subgroup of | #72233511, #72233542, #72233544, … |
| maximal quotient of | #72261729, #72262801, #72297559, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8138891);