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 | |
5982 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,4,6)(3,5,7,8)(9,10,13,14)(11,15,16,12)(17,18,22,23)(19,20,26,24)(21,29,30,25)(27,31,32,28);(1,3,4,7)(2,5,6,8)(10,14)(12,15)(18,23)(20,24)(21,30)(27,32);(1,2)(3,5)(4,6)(7,8)(9,10)(11,12)(13,14)(15,16)(18,23)(19,26)(25,29)(27,32);(9,11)(10,12)(13,16)(14,15)(17,19,22,26)(18,24,23,20)(21,27,30,32)(25,31,29,28);(1,3)(2,5)(4,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,20,22,24)(18,19,23,26)(21,28,30,31)(25,27,29,32);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,21)(18,25)(19,27)(20,28)(22,30)(23,29)(24,31)(26,32)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=gi,e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi83 conjugacy classes — view the full character table.
| maximal subgroup of | #72261815, #72261948, #72262018, … |
| maximal normal subgroup of | #72261815, #72261948, #72262018, … |
| maximal quotient of | #72300520, #72300543 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8769575);