Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C22 | C2×D42 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(6,12,15,24)(7,16,19,18)(8,20)(14,26,30,21)(17,31,34,33)(22,23)(27,36)(32,35);(1,2)(3,4)(5,6)(7,13)(8,21)(9,12)(10,24)(11,15)(14,22)(16,28)(17,32)(18,25)(19,29)(20,26)(23,30)(27,33)(31,36)(34,35);(2,4)(5,7)(6,13)(8,17)(9,18)(10,16)(11,19)(12,25)(14,27)(15,29)(20,34)(21,32)(22,33)(23,31)(24,28)(26,35)(30,36);(5,8)(6,14)(7,17)(9,22)(10,23)(11,20)(12,26)(13,27)(15,30)(16,31)(18,33)(19,34)(21,24)(25,35)(28,32)(29,36);(5,9,11,10)(6,12,15,24)(7,18,19,16)(8,22,20,23)(13,25,29,28)(14,26,30,21)(17,33,34,31)(27,35,36,32)Polycyclic presentation:
2^9|a^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,f^b=fi,g^c=gi,f^d=fi,g^d=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410562);