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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,26,30,33)(27,31,36,32)(28,29,38,40)(34,35,37,39);(2,5)(6,7)(10,14)(15,18)(16,21)(22,24)(25,27)(26,31)(28,34)(29,35)(30,36)(32,33)(37,38)(39,40);(1,2)(3,6)(4,5)(7,8)(11,17)(12,20)(15,18)(16,21)(25,27,30,36)(26,32,33,31)(28,34,38,37)(29,39,40,35);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(25,27)(26,32)(28,37)(29,35)(30,36)(31,33)(34,38)(39,40);(9,10)(11,18)(12,16)(13,14)(15,17)(19,24)(20,21)(22,23)(25,28)(26,29)(27,34)(30,38)(31,35)(32,39)(33,40)(36,37);(9,12)(10,16)(11,19)(13,20)(14,21)(15,22)(17,23)(18,24)(25,29)(26,28)(27,35)(30,40)(31,34)(32,37)(33,38)(36,39)Polycyclic presentation:
2^9|a^2=h,c^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72333431, #72341026, #72341035, … |
| maximal normal subgroup of | #72333431, #72341026, #72341035, … |
| maximal quotient of | #72333446, #72333450, #72333476, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8834233);