Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393944 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14)(15,17)(16,18)(19,20)(21,22,25,28)(23,31,34,35)(24,29,26,30)(27,33,36,32);(2,4)(13,15,16,19)(14,17,18,20)(21,23)(22,27)(24,33)(25,34)(26,32)(28,36)(29,31)(30,35);(5,7,8,12)(6,9,10,11)(22,28)(23,32)(24,26)(27,35)(31,36)(33,34);(14,18)(17,20)(21,24,25,26)(22,29,28,30)(23,33,34,32)(27,31,36,35)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=fi,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,d^c=di,e^c=ei,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313942);