Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994043 |
| 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 | C2×#553 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#124 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1736 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)(5,6,9,13)(7,11,14,10)(8,16,18,12)(15,19,17,20)(21,22)(23,29)(24,33)(25,27)(26,32)(28,34)(30,36)(31,35);(5,7,9,14)(6,10,13,11)(8,17,18,15)(12,20,16,19)(21,23,25,30)(22,26,27,35)(24,31,34,32)(28,36,33,29);(1,2,3,4)(5,7)(6,11)(8,15)(9,14)(10,13)(12,20)(16,19)(17,18)(22,27)(23,30)(28,33)(31,32);(7,14)(10,11)(15,17)(19,20)(22,27)(23,31)(24,34)(26,29)(30,32)(35,36);(5,8,9,18)(6,12,13,16)(7,15,14,17)(10,19,11,20)(22,27)(23,30)(28,33)(31,32)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7100354);