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 | |
4528 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(2,4)(6,8)(9,10)(11,14)(12,17)(13,16)(15,20)(18,21)(19,23)(22,24)(26,31)(28,35)(29,37)(34,39);(3,7)(6,8)(10,11)(15,17)(16,18)(22,23)(26,31)(27,33)(28,35)(36,40);(1,2,5,4)(3,6,7,8)(9,10,14,11)(12,15,20,17)(13,16,21,18)(19,22,24,23)(25,26)(27,29)(28,32)(30,31)(33,37)(34,36)(35,38)(39,40);(1,3)(2,6)(4,8)(5,7)(9,11,14,10)(12,17,20,15)(13,18,21,16)(19,23,24,22)(25,27)(26,29)(28,34)(30,33)(31,37)(32,36)(35,39)(38,40);(1,4,5,2)(3,8,7,6)(9,12)(10,15)(11,17)(13,19)(14,20)(16,22)(18,23)(21,24)(25,28,30,35)(26,32,31,38)(27,34,33,39)(29,36,37,40);(9,13,14,21)(10,16,11,18)(12,19,20,24)(15,22,17,23)(25,29)(26,27)(28,36)(30,37)(31,33)(32,34)(35,40)(38,39)Polycyclic presentation:
2^9|c^2=ghi,d^2=g,e^2=gh,f^2=g,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 subgroup of | #72219287 |
| maximal normal subgroup of | #72219281, #72219283, #72219287 |
| maximal quotient of | #72219274 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8015905);