Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C22×D4 | C22×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
659 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,13)(8,17)(9,11)(10,16)(12,18)(14,19)(15,20)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(5,7,9,14)(6,10,11,20)(8,15,18,16)(12,19,17,13)(23,29)(26,32)(30,35)(33,36);(1,2,3,4)(6,11)(7,14)(12,17)(15,16)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(7,15)(8,18)(10,19)(12,17)(13,20)(14,16)(23,29)(26,32)(30,35)(33,36)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=fh,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=chi,d^b=de,e^b=eh,d^c=di,e^d=eh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 132666);