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 | |
6052 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,25)(20,22)(21,24)(23,26)(27,31)(28,29)(30,32);(3,7)(6,8)(10,14)(11,16)(18,20)(21,29)(23,25)(27,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(19,26)(23,25)(27,32)(30,31);(9,12)(10,11)(13,15)(14,16)(17,19)(18,23)(20,25)(21,27)(22,26)(24,30)(28,31)(29,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,20,22,18)(19,25,26,23)(21,28,29,24)(27,31,32,30);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)Polycyclic presentation:
2^9|e^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261907, #72262018, #72319702, … |
| maximal normal subgroup of | #72261907, #72262018, #72319702, … |
| maximal quotient of | #72300491 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8662653);