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 | |
8052 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)(7,9)(8,10)(11,12)(14,16)(18,19);(1,2)(3,4)(5,6,9,7)(8,11,12,10)(13,14,17,16)(15,18,20,19);(6,7)(10,11)(13,14,17,16)(15,19,20,18);(5,6,9,7)(8,10,12,11)(13,15,17,20)(14,18,16,19);(5,7,9,6)(8,10,12,11)(13,16,17,14)(15,19,20,18);(5,8)(6,10)(7,11)(9,12)Polycyclic presentation:
2^9|b^2=h,c^2=hi,d^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 | #72262454, #72262464, #72262490, … |
| maximal normal subgroup of | #72262454, #72262464, #72262490, … |
| maximal quotient of | #72281305, #72300348, #72300361, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7879938);