Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1046 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)(7,11)(8,10)(9,12)(13,14)(15,18)(16,17)(19,20)(21,22)(23,30)(24,33)(25,29)(26,28)(27,32)(31,35)(34,36);(5,7,8,12)(6,9,10,11)(21,23,24,34,25,32,26,31)(22,27,33,35,29,30,28,36);(14,17)(18,20)(21,23,25,32)(22,27,29,30)(24,31,26,34)(28,35,33,36);(6,10)(9,11)(13,15,16,19)(14,18,17,20)(21,24,25,26)(22,28,29,33)(23,31,32,34)(27,35,30,36)Polycyclic presentation:
2^9|a^2=h,b^2=eg,c^2=i,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 269575);