Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6346727 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3497 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,6)(3,4)(7,8)(9,10)(11,16)(12,18)(13,17)(14,19)(15,20)(21,23)(22,24);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,20)(13,21)(14,22)(16,17)(18,24)(19,23);(3,4)(6,8)(9,11)(10,16)(12,21)(13,20)(14,22)(15,17)(18,23)(19,24);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,17,19,18)(11,20,22,21)(15,16,23,24);(2,7)(6,8)(9,12,14,13)(10,18,19,17)(11,21,22,20)(15,16,23,24)Polycyclic presentation:
2^9|d^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72307970, C2×#6346754 |
| maximal normal subgroup of | #72307970, C2×#6346754 |
| maximal quotient of | C2×#6346754, #72309172, #72309174, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6253950);