Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7501009 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3533 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,17,21,27)(8,22,24,30)(9,15)(10,16)(11,14)(12,20,28,19)(13,23,29,25)(18,26)(31,33)(32,35)(34,36);(1,2)(3,4)(5,7)(6,12)(8,18)(9,19)(10,20)(11,21)(13,26)(14,28)(15,27)(16,17)(22,35)(23,32)(24,34)(25,33)(29,36)(30,31);(2,4)(5,8)(6,13)(7,18)(9,23)(10,25)(11,24)(12,26)(14,29)(15,22)(16,30)(17,31)(19,32)(20,33)(21,34)(27,35)(28,36);(7,19)(9,10)(12,27)(15,16)(17,28)(18,32)(20,21)(22,30)(23,25)(26,35)(31,36)(33,34);(6,14)(12,28)(13,29)(15,16)(17,27)(22,30)(26,36)(31,35)Polycyclic presentation:
2^9|a^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,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, 7410563);