Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393940 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1214 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)(5,6,8,10)(7,11,12,9)(13,14,16,17)(15,19,20,18)(21,22,26,30)(23,31,34,35)(24,29,25,28)(27,32,36,33);(5,7,8,12)(6,9,10,11)(14,17)(18,19)(23,32)(24,25)(27,35)(28,29)(31,36)(33,34);(2,4)(21,23,26,34)(22,27,30,36)(24,32,25,33)(28,35,29,31);(13,15,16,20)(14,18,17,19)Polycyclic presentation:
2^9|a^2=ei,b^2=eh,c^2=i,d^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 323198);