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 | |
7896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,3)(4,8)(9,10,13,15)(11,14,16,12)(18,21)(22,24);(1,2,5,3)(4,6,8,7)(9,11,13,16)(10,14,15,12)(19,23)(22,24);(1,2,5,3)(4,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,18)(19,22)(20,21)(23,24);(2,3)(6,7)(10,15)(11,16);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,19,20,23)(18,22,21,24);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,18)(19,22)(20,21)(23,24)Polycyclic presentation:
2^9|a^2=i,b^2=gi,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=egi,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219532, #72232277, #72232563, … |
| maximal normal subgroup of | #72219532, #72232277, #72232563, … |
| maximal quotient of | #72219499, #72222747, #72222757, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8841573);