Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C2×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C4⋊C4 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
838 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)(7,14)(8,12)(9,18)(10,13)(11,17)(15,20)(16,19)(21,22,24,27,25,28,31,34)(23,26,30,33,29,32,35,36);(5,7,9,17,10,15,8,16)(6,11,12,14,13,19,18,20)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(1,2,3,4)(7,15)(11,19)(14,20)(16,17)(23,29)(26,32)(30,35)(33,36);(5,8,10,9)(6,12,13,18)(7,16,15,17)(11,14,19,20)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=f,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,c^b=chi128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 98576);