Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
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 | |
9124 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(5,6)(7,8)(9,11)(10,12)(14,17)(15,19);(1,2)(3,4)(14,17)(18,20);(5,7)(6,10)(8,11)(9,12)(13,14,16,17)(15,18,19,20);(5,8,9,10)(6,7,11,12)(13,15,16,19)(14,18,17,20);(5,6)(7,8)(9,11)(10,12);(5,7)(6,10)(8,11)(9,12)Polycyclic presentation:
2^9|c^2=i,d^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fhi,c^b=ci,e^c=ehi,f^e=fhi110 conjugacy classes — view the full character table.
| maximal subgroup of | #72233744, #72233768, #72234453, … |
| maximal normal subgroup of | #72233744, #72233768, #72234453, … |
| maximal quotient of | #72261594, #72262164, #72263126, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7572134);