Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61559 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C23×C22⋊C4 |
| prime radical | #412 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
3299 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8,6,3,4,7)(10,15)(12,13)(16,22)(17,21)(19,20)(23,24);(1,3)(2,6)(4,8)(5,7)(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,22)(20,24);(9,11)(10,16)(12,19)(13,20)(14,18)(15,22)(17,23)(21,24);(11,18)(16,23)(19,20)(22,24);(1,4,6,5)(2,7,3,8)(9,12,14,13)(10,15,17,21)(11,19,18,20)(16,22,23,24)Polycyclic presentation:
2^9|a^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,e^a=ehi,f^a=fh,f^b=fh,d^c=dhi122 conjugacy classes — view the full character table.
| maximal subgroup of | #72338446, #72338447, #72338450 |
| maximal normal subgroup of | #72338446, #72338447, #72338450 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250334);