Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #39491 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1842 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22,25,29)(23,31,33,27)(24,34,35,28)(26,30,32,36);(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(23,24)(26,32)(27,28)(30,36)(31,34)(33,35);(1,2,3,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,23)(22,27)(24,32)(25,33)(26,35)(28,36)(29,31)(30,34);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19);(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,24,25,35)(22,28,29,34)(23,32,33,26)(27,36,31,30)Polycyclic presentation:
2^9|a^2=i,c^2=fgi,d^2=f,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,h^c=hi,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594335);