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 | |
5350 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,4)(5,7)(6,8)(9,10)(11,12)(13,14)(15,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(1,3)(2,6)(4,7)(5,8)(10,14)(11,15)(19,25)(23,29)(26,31)(30,32);(9,11,13,15)(10,12,14,16)(18,22)(23,29)(24,27)(30,32);(9,12)(10,15)(11,14)(13,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(1,4,5,6)(2,3,7,8)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,3)(2,4)(5,8)(6,7)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)Polycyclic presentation:
2^9|c^2=i,e^2=gi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi83 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219217, #72219279, #72219281 |
| maximal quotient of | #72219216 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 9626412);