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 | |
5687 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,4)(3,5)(10,13)(15,16)(17,18)(19,24)(20,29)(21,31)(22,25)(23,26)(27,28)(30,32);(1,2,6,4)(3,8,5,7)(9,10,14,13)(11,15,12,16)(19,26)(23,24)(27,30)(28,32);(1,2,6,4)(3,8,5,7)(9,11,14,12)(10,15,13,16)(17,19,22,26)(18,23,25,24)(20,27,31,30)(21,28,29,32);(1,3,6,5)(2,7,4,8)(9,10,14,13)(11,16,12,15)(17,19)(18,24)(20,30)(21,28)(22,26)(23,25)(27,31)(29,32);(1,4,6,2)(3,7,5,8)(9,12,14,11)(10,16,13,15)(17,20)(18,21)(19,27)(22,31)(23,28)(24,32)(25,29)(26,30);(1,5,6,3)(2,7,4,8)(9,13,14,10)(11,16,12,15)(17,21)(18,20)(19,28)(22,29)(23,27)(24,30)(25,31)(26,32)Polycyclic presentation:
2^9|b^2=g,c^2=gi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,e^d=egi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72312548, #72342303, #72342310, … |
| maximal normal subgroup of | #72312548, #72342303, #72342310, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8064015);