Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
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 | |
8832 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)(5,6,9,8)(7,11,12,10)(13,14)(15,19)(16,17)(18,20);(1,2,3,4)(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,20)(17,18);(7,12)(10,11)(14,20)(15,17);(5,6,9,8)(7,11,12,10)(13,15,18,17)(14,19,20,16);(5,8,9,6)(7,10,12,11)(13,16)(14,15)(17,20)(18,19);(5,7)(6,11)(8,10)(9,12)(13,17,18,15)(14,16,20,19)Polycyclic presentation:
2^9|a^2=hi,b^2=g,d^2=i,e^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72234860, #72234875, #72282768 |
| maximal normal subgroup of | #72234860, #72234875, #72282768 |
| maximal quotient of | #72261555, #72262099 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713680);