Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454382 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,22,25,29,26,30,24,28)(23,27,34,36,32,35,33,31);(6,9)(10,12)(13,15,16,19)(14,17,18,20)(21,23,25,34,26,32,24,33)(22,27,29,36,30,35,28,31);(7,11)(10,12)(14,18)(17,20)(21,22,26,30)(23,31,32,36)(24,29,25,28)(27,34,35,33);(2,4)(5,7,8,11)(6,10,9,12)(23,32)(27,35)(31,36)(33,34)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=i,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367941);