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 | |
3431 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,17,20,32)(8,13)(9,15)(10,16)(11,14)(12,19,27,25)(18,28,34,33)(21,36,35,26)(22,30)(23,31)(24,29);(1,2)(3,4)(5,7)(6,12)(8,21)(9,25)(10,19)(11,20)(13,28)(14,27)(15,32)(16,17)(18,22)(23,34)(24,35)(26,30)(29,33)(31,36);(2,4)(5,7)(6,12)(8,18)(9,19)(10,25)(11,20)(13,26)(14,27)(15,17)(16,32)(21,22)(23,35)(24,34)(28,30)(29,36)(31,33);(5,8,11,24)(6,13,14,29)(7,18,20,34)(9,23,10,22)(12,26,27,36)(15,31,16,30)(17,33,32,28)(19,35,25,21);(6,14)(12,27)(13,29)(15,16)(17,32)(26,36)(28,33)(30,31)Polycyclic presentation:
2^9|a^2=fg,d^2=i,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, 7410564);