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 | |
5250 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,6,8)(3,7,5,4)(9,10)(11,15)(12,16)(13,14)(17,18,22,23)(19,25,26,24)(20,21,29,31)(27,28,30,32);(1,3,6,5)(2,7,8,4)(10,14)(11,16)(18,23)(20,29)(24,25)(27,30);(2,8)(3,5)(9,11)(10,12)(13,16)(14,15)(18,23)(19,26)(21,31)(27,30);(1,3,6,5)(2,7,8,4)(9,12)(10,11)(13,15)(14,16)(17,19)(18,24)(20,30)(21,28)(22,26)(23,25)(27,29)(31,32);(1,4,6,7)(2,3,8,5)(9,10)(11,12)(13,14)(15,16)(17,20)(18,21)(19,27)(22,29)(23,31)(24,32)(25,28)(26,30);(1,5,6,3)(2,7,8,4)(9,12)(10,11)(13,15)(14,16)(17,21)(18,20)(19,28)(22,31)(23,29)(24,30)(25,27)(26,32)Polycyclic presentation:
2^9|a^2=gi,b^2=gh,d^2=gh,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72265955, #72265959, #72265962, … |
| maximal normal subgroup of | #72265955, #72265959, #72265962, … |
| maximal quotient of | #72266508, #72266558, #72300635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8815231);