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 | |
9460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(13,14,17,18)(15,16,19,20);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10);(6,10)(8,11)(14,18)(15,19);(5,7)(6,8)(9,12)(10,11)(13,15)(14,16)(17,19)(18,20);(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19);(5,8)(6,7)(9,11)(10,12)Polycyclic presentation:
2^9|a^2=gh,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 | #72232293, #72234912, #72234914, … |
| maximal normal subgroup of | #72232293, #72234912, #72234914, … |
| maximal quotient of | #72232353, #72232387, #72260748, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7711625);