Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393889 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1448 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,17)(16,18)(19,20)(21,22)(23,28)(24,29)(25,34)(26,30)(27,31)(32,35)(33,36);(1,2)(3,4)(13,15,16,19)(14,17,18,20)(21,23)(22,27)(24,31)(25,33)(26,32)(28,34)(29,35)(30,36);(2,4)(5,7,8,12)(6,9,10,11)(21,23)(22,28)(24,33)(25,31)(26,32)(27,34)(29,36)(30,35);(14,18)(17,20)(21,24,26,25)(22,29,30,34)(23,31,32,33)(27,35,36,28)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=f,d^2=i,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 313913);