Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#127 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1898 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2,4,3)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,29)(23,27,25,34)(24,31,32,36)(28,30,35,33);(2,3)(7,13)(8,16)(10,17)(11,15)(21,23,26,25)(22,27,29,34)(24,30,32,33)(28,31,35,36);(5,7,9,13)(6,10,12,17)(8,14,16,19)(11,18,15,20)(21,24,25,33,26,32,23,30)(22,28,27,31,29,35,34,36);(1,3,4,2)(7,13)(10,17)(14,19)(18,20);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(24,32)(28,35)(30,33)(31,36)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=gh,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7104046);