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 | |
6709 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)(3,7)(4,6)(5,8)(9,10,13,15)(11,16,12,14)(17,18,22,23)(19,20,24,21);(1,3)(2,5)(4,8)(6,7)(11,12)(14,16)(17,19)(18,21)(20,23)(22,24);(9,11,13,12)(10,14,15,16)(17,20,22,21)(18,19,23,24);(2,6)(5,7)(11,12)(14,16)(18,23)(20,21);(1,2)(3,5)(4,6)(7,8)(9,12,13,11)(10,14,15,16)(17,21,22,20)(18,19,23,24);(1,3)(2,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,19)(18,21)(20,23)(22,24)Polycyclic presentation:
2^9|a^2=h,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72232575, #72262652, #72263266, … |
| maximal normal subgroup of | #72232575, #72262652, #72263266, … |
| maximal quotient of | #72232668, #72262680, #72327365, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9994912);