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 | |
6262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(3,7)(11,15)(13,16)(17,18)(19,24)(20,28)(21,26)(22,23)(25,30)(27,31)(29,32);(1,2,4,5)(3,6,7,8)(9,10,12,14)(11,13,15,16)(18,23)(20,30)(24,32)(26,27);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,15)(14,16)(18,23)(21,31)(24,32)(25,28);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,19)(18,24)(20,26)(21,28)(22,29)(23,32)(25,31)(27,30);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,15)(14,16)(17,20)(18,25)(19,27)(21,24)(22,30)(23,28)(26,29)(31,32);(17,21)(18,26)(19,28)(20,24)(22,31)(23,27)(25,29)(30,32)Polycyclic presentation:
2^9|b^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72262081, #72262208, #72326660, … |
| maximal normal subgroup of | #72262081, #72262208, #72326660, … |
| maximal quotient of | #72326701, #72326829, #72327219, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8313699);