Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #35520 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(21,22)(23,32)(24,34)(25,30)(26,36)(27,31)(28,33)(29,35);(2,4)(5,6)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,23,27,24)(22,28,31,29)(25,33,26,35)(30,34,36,32);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,24,27,23)(22,29,31,28)(25,35,26,33)(30,32,36,34);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(21,22)(23,28)(24,29)(25,36)(26,30)(27,31)(32,33)(34,35);(6,12)(10,13)(11,20)(18,19)(21,25,27,26)(22,30,31,36)(23,33,24,35)(28,34,29,32)Polycyclic presentation:
2^9|b^2=i,c^2=hi,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6436380);