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 | |
7244 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,5,8)(3,4,7,6)(10,14)(11,16)(17,18)(19,23)(20,24)(21,22);(1,3,5,7)(2,4,8,6)(9,10)(11,15)(12,16)(13,14)(18,22)(19,24);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,14)(13,16)(17,19)(18,20)(21,24)(22,23);(1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,24)(22,23);(17,20)(18,19)(21,23)(22,24);(1,4)(2,7)(3,8)(5,6)(9,12)(10,16)(11,14)(13,15)Polycyclic presentation:
2^9|a^2=gh,b^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72234855, #72234862, #72234867, … |
| maximal normal subgroup of | #72234855, #72234862, #72234867, … |
| maximal quotient of | #72261083, #72261160, #72261226, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8336968);