Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | #59041 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4600 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(13,21)(16,20)(18,24)(22,23)(25,26,31,36)(27,37,38,32)(28,33,34,29)(30,40,39,35);(2,3)(6,7)(10,12)(15,17)(16,20)(22,23)(27,38)(28,34)(29,33)(32,37);(1,2,5,3)(4,6,8,7)(9,10,14,12)(11,15,19,17)(13,16,21,20)(18,22,24,23)(25,27)(26,32)(28,39)(29,35)(30,34)(31,38)(33,40)(36,37);(1,3,5,2)(4,7,8,6)(9,11,14,19)(10,15,12,17)(13,18,21,24)(16,22,20,23)(25,28,31,34)(26,33,36,29)(27,39,38,30)(32,40,37,35);(1,4,5,8)(2,6,3,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,29,31,33)(26,34,36,28)(27,35,38,40)(30,37,39,32);(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,30,31,39)(26,35,36,40)(27,34,38,28)(29,37,33,32)Polycyclic presentation:
2^9|a^2=gh,c^2=ghi,d^2=i,e^2=i,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^b=egi92 conjugacy classes — view the full character table.
| maximal normal subgroup of | #72219283, #72219285 |
| maximal quotient of | #72219274 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8017809);