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 | |
10584 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)(13,14)(15,20)(16,17)(18,19);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,14)(15,20)(16,17)(18,19);(6,10)(8,11)(14,19)(15,17);(13,15,18,17)(14,20,19,16);(5,7)(6,11)(8,10)(9,12)(13,16)(14,15)(17,19)(18,20);(5,8,9,11)(6,7,10,12)(13,17,18,15)(14,16,19,20)Polycyclic presentation:
2^9|d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72233746, #72233751, #72234449, … |
| maximal normal subgroup of | #72233746, #72233751, #72234449, … |
| maximal quotient of | #72261246, #72261313, #72261404, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7653909);