Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4932388 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4715 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)(3,4)(5,6,7,8)(9,10,13,11)(12,15,16,14)(17,18)(19,21)(20,22)(23,24);(9,10,13,11)(12,14,16,15)(18,19)(20,24);(2,3)(10,11)(14,15)(17,19,21,18)(20,23,24,22);(1,3,4,2)(5,6)(7,8)(9,11,13,10)(12,15,16,14);(9,12)(10,14)(11,15)(13,16)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=fgi,b^2=fh,c^2=f,d^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg80 conjugacy classes — view the full character table.
| maximal subgroup of | #72297962, #72297972, #72297973, … |
| maximal normal subgroup of | #72297962, #72297972, #72297973, … |
| maximal quotient of | #72298350, #72298351 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4848498);