Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
4415 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)(11,15)(12,16)(13,14)(18,22)(19,23);(1,2)(3,4)(5,6,7,8)(9,11)(10,12)(13,16)(14,15)(17,18,21,22)(19,24,23,20);(1,2,3,4)(5,6)(7,8)(10,14)(12,15)(18,22)(20,24);(9,11)(10,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,24);(1,2)(3,4)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,24)(22,23)Polycyclic presentation:
2^9|b^2=fhi,c^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72300426, #72300431, #72300453, … |
| maximal normal subgroup of | #72300426, #72300431, #72300453, … |
| maximal quotient of | #72330548, #72330934 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2161967);