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 | |
9993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,20)(16,19)(17,18);(1,2)(3,4)(6,9)(10,11)(13,14,18,17)(15,20,19,16);(5,6)(7,10)(8,9)(11,12)(13,15,18,19)(14,16,17,20);(5,7)(6,10)(8,12)(9,11)(13,16)(14,15)(17,19)(18,20);(5,7)(6,11)(8,12)(9,10)(13,17,18,14)(15,16,19,20);(5,6)(7,10)(8,9)(11,12)(13,16)(14,19)(15,17)(18,20)Polycyclic presentation:
2^9|b^2=h,c^2=h,e^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72232031, #72232070, #72232817, … |
| maximal normal subgroup of | #72232031, #72232070, #72232817, … |
| maximal quotient of | #72232060, #72232095, #72232123, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7874022);