Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C210×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #35346 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6652 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)(9,10,13,15)(11,12,16,14)(20,27)(24,29)(26,31)(30,32);(2,5)(3,7)(9,11,13,16)(10,14,15,12)(18,22)(19,25)(24,29)(26,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,14)(15,16);(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72249516, #72249541, #72258185, … |
| maximal normal subgroup of | #72249516, #72249541, #72258185, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8000368);