Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23924 |
| 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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1008 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(21,22,24,28)(23,30,31,26)(25,27,33,29)(32,35,36,34);(1,3,4,2)(6,9)(10,11)(13,14,16,18)(15,19,20,17)(21,23)(22,26)(24,31)(25,32)(27,34)(28,30)(29,35)(33,36);(13,15,16,20)(14,17,18,19)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36);(5,7,8,12)(6,10,9,11)(22,28)(23,31)(27,29)(32,36)Polycyclic presentation:
2^9|a^2=h,b^2=ghi,c^2=g,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,f^c=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 277218);