Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×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 | |
5318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,14)(12,16)(13,15)(18,22)(19,20);(5,6,7,8)(9,11,13,16)(10,14,15,12)(17,18)(19,23)(20,24)(21,22);(5,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,20)(18,23,22,24);(1,2)(3,4)(9,11)(10,12)(13,16)(14,15)(17,20,21,19)(18,24,22,23);(9,12)(10,11)(13,14)(15,16)(17,18)(19,24)(20,23)(21,22)Polycyclic presentation:
2^9|b^2=h,c^2=f,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72232377, #72232391, #72232406, … |
| maximal normal subgroup of | #72232377, #72232391, #72232406, … |
| maximal quotient of | #72293936, #72293968, C2×#3334777, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3241973);