Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24384 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
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 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1178 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)(21,22)(23,29)(24,26)(25,33)(27,31)(28,32)(30,35)(34,36);(5,7,8,12)(6,9,10,11)(13,14)(15,18)(16,17)(19,20)(22,26)(27,34)(28,33)(29,35);(2,4)(14,17)(18,20)(21,23,25,31,24,30,32,36)(22,27,28,29,26,34,33,35);(6,10)(9,11)(13,15,16,19)(14,18,17,20)(22,26)(23,30)(28,33)(31,36)Polycyclic presentation:
2^9|a^2=h,b^2=ei,c^2=fh,d^2=g,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 282837);