Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481415 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #61433 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C8×#41 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
735 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(18,22,23,30)(19,25)(20,28)(24,26,29,31)(27,32);(17,18,21,23)(19,26,27,31)(20,29,28,24)(22,25,30,32);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16);(3,9)(6,12)(10,15)(13,16)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=g,f^2=h,g^2=i,b^a=be,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285760, #72287682, #72287706, … |
| maximal normal subgroup of | #72285760, #72287682, #72287706, … |
| maximal quotient of | #72286006, #72286976, #72343615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388614);