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 | |
5878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,23)(22,25)(27,32)(28,31)(29,30);(1,3)(2,4)(5,8)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32);(1,3,5,8)(2,6,7,4)(10,14)(13,15)(17,20,23,19)(18,24,21,26)(22,27,30,28)(25,32,29,31);(9,11)(10,13)(12,16)(14,15)(17,21,23,18)(19,24,20,26)(22,29,30,25)(27,31,28,32);(1,4)(2,3)(5,6)(7,8)(9,11)(10,13)(12,16)(14,15);(1,3)(2,4)(5,8)(6,7)(9,10)(11,13)(12,14)(15,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)Polycyclic presentation:
2^9|a^2=i,b^2=ghi,c^2=gh,d^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72210417 |
| maximal normal subgroup of | #72210417 |
| maximal quotient of | #72210242, #72266747, #72266759, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8096600);