Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453981 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(14,17)(15,19)(21,22)(23,31)(24,29)(25,28)(26,30)(27,34)(32,35)(33,36);(13,14,16,17)(15,18,19,20)(21,23,25,34)(22,27,28,31)(24,33,26,32)(29,35,30,36);(5,7,8,12)(6,9,10,11)(13,15,16,19)(14,18,17,20)(22,28)(23,32)(24,26)(27,35)(31,36)(33,34);(2,4)(15,19)(18,20)(21,24,25,26)(22,29,28,30)(23,33,34,32)(27,35,31,36)Polycyclic presentation:
2^9|a^2=ef,b^2=e,c^2=fi,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,d^c=de,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 364898);