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 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,17,20)(15,18,16,19)(21,22,24,26)(23,27,28,25);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(21,22)(23,25)(24,26)(27,28);(1,2,3,4)(7,8)(10,12)(13,15,17,16)(14,18,20,19)(21,23,24,28)(22,25,26,27);(5,6)(7,12)(8,10)(9,11)(13,14)(15,18)(16,19)(17,20)(21,22)(23,25)(24,26)(27,28);(1,2)(3,4)(5,8,9,7)(6,10,11,12)(13,16,17,15)(14,19,20,18)(21,23)(22,25)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=fghi,e^2=ghi,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 | #72326695, #72326698, #72326773, … |
| maximal normal subgroup of | #72326695, #72326698, #72326773, … |
| maximal quotient of | #72330278, #72330863 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2450143);