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 | |
4964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,5,7)(3,4,8,6)(9,10)(11,17)(12,16)(13,14)(15,18)(19,24)(20,22)(21,23)(25,26)(27,31)(28,36)(29,30)(32,37)(33,35)(34,39)(38,40);(1,3,5,8)(2,6,7,4)(10,14)(11,18)(16,22)(19,21)(25,26)(27,33)(28,36)(29,30)(31,35)(32,37)(34,38)(39,40);(1,3)(2,4)(5,8)(6,7)(12,20)(16,22)(19,21)(23,24)(26,30)(31,33)(32,36)(38,39);(1,4)(2,3)(5,6)(7,8)(9,11)(10,15)(12,21)(13,18)(14,17)(16,24)(19,20)(22,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,39)(37,40);(9,12)(10,16)(11,19)(13,20)(14,22)(15,23)(17,24)(18,21)(25,26)(27,31)(28,32)(29,30)(33,35)(34,38)(36,37)(39,40);(9,10)(11,15)(12,16)(13,14)(17,18)(19,23)(20,22)(21,24)(25,28)(26,32)(27,34)(29,37)(30,36)(31,38)(33,39)(35,40)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi83 conjugacy classes — view the full character table.
| Fitting subgroup of | #72340966 |
| maximal subgroup of | #72340966, #72340970 |
| maximal normal subgroup of | #72340966, #72340970 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10239360);