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 | |
6238 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)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,24)(20,21)(22,23)(25,31)(26,32)(27,29)(28,30);(3,7)(6,8)(10,14)(11,16)(18,23)(20,28)(24,31)(26,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(19,25)(20,28)(21,30)(24,31);(9,12)(10,11)(13,15)(14,16)(17,19)(18,24)(20,29)(21,27)(22,25)(23,31)(26,28)(30,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,12)(13,14)(15,16)(17,20)(18,21)(19,26)(22,28)(23,30)(24,32)(25,29)(27,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,20)(19,27)(22,30)(23,28)(24,29)(25,32)(26,31)Polycyclic presentation:
2^9|b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72261997, #72262081, #72326785, … |
| maximal normal subgroup of | #72261997, #72262081, #72326785, … |
| maximal quotient of | #72326797, #72326799, #72326880, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8948218);