Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417442 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1152 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)(6,9)(10,12)(13,14,16,17)(15,18,19,20)(21,22,25,28)(23,26,31,35)(24,27,32,36)(29,33,30,34);(5,6,8,9)(7,10,11,12)(14,17)(15,19)(21,23,25,31)(22,26,28,35)(24,29,32,30)(27,33,36,34);(5,7,8,11)(6,10,9,12)(14,17)(18,20)(23,29)(24,32)(26,33)(27,36)(30,31)(34,35);(2,4)(6,9)(10,12)(13,15,16,19)(14,18,17,20)Polycyclic presentation:
2^9|a^2=fi,b^2=efi,c^2=ef,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,g^b=gi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 339554);