Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213564 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1153 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,10)(8,15,16,11)(9,12)(13,19,20,17)(14,18)(21,22,24,28,25,27,33,35)(23,29,31,36,30,26,32,34);(1,2)(3,4)(5,7,9,14)(6,10,12,18)(8,13,16,20)(11,17,15,19)(23,30)(26,29)(31,32)(34,36);(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,20)(18,19)(21,23)(22,26)(24,32)(25,30)(27,29)(28,36)(31,33)(34,35);(2,4)(22,27)(23,30)(28,35)(31,32)Polycyclic presentation:
2^9|a^2=g,b^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,c^b=ci,d^b=de,d^c=di,g^c=gh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 126979);