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 | |
6312 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)(9,10,13,14)(11,15,16,12)(17,18)(19,23)(20,30)(21,25)(22,26)(24,28)(27,29)(31,32);(3,7)(6,8)(10,14)(12,15)(19,27)(21,31)(23,29)(25,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,27)(18,23,26,29)(20,21,28,31)(24,32,30,25);(9,11)(10,12)(13,16)(14,15)(17,20)(18,24)(19,21)(22,28)(23,32)(25,29)(26,30)(27,31);(1,3)(2,6)(4,7)(5,8)(9,12)(10,11)(13,15)(14,16)(17,21)(18,25)(19,28)(20,27)(22,31)(23,24)(26,32)(29,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,29)(20,24)(21,25)(22,26)(23,27)(28,30)(31,32)Polycyclic presentation:
2^9|a^2=g,c^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72235981, #72240624, #72252234, … |
| maximal normal subgroup of | #72235981, #72240624, #72252234, … |
| maximal quotient of | #72235982, #72283045, #72326855, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8833865);