Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| 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 | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3583 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,9,10)(7,12,11,8)(13,14,17,15)(16,19,20,18)(22,24)(23,27);(5,6,9,10)(7,8,11,12)(14,15)(18,19)(21,22,25,24)(23,28,27,26);(2,3)(6,10)(7,11)(13,14,17,15)(16,18,20,19)(21,23,25,27)(22,26,24,28);(1,3,4,2)(5,7,9,11)(6,8,10,12)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,28)(25,27);(5,8)(6,11)(7,10)(9,12)(13,16,17,20)(14,18,15,19)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=f,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72326556, #72326594, #72326629, … |
| maximal normal subgroup of | #72326556, #72326594, #72326629, … |
| maximal quotient of | #72328998 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836467);