Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2031997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15267 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3730 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)(8,11)(9,10)(13,14,17,18)(15,19,20,16)(22,25)(23,27);(5,7,10,8)(6,11,9,12)(13,15,17,20)(14,16,18,19)(23,27)(26,28);(5,7,10,8)(6,12,9,11)(13,15)(14,16)(17,20)(18,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,26)(24,27)(25,28);(5,9,10,6)(7,11,8,12)(13,16)(14,15)(17,19)(18,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=fh,d^2=fh,e^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72300846, #72300876, #72300881, … |
| maximal normal subgroup of | #72300846, #72300876, #72300881, … |
| maximal quotient of | #72329833 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1947110);