Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1082705 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4996 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(6,8)(13,14,18,21)(15,22,19,16)(17,25,27,20)(23,24,26,28);(10,12)(13,15,18,19)(14,16,21,22)(17,26,27,23)(20,28,25,24);(1,2)(3,4)(5,6)(7,8)(9,10,11,12)(13,14)(15,16)(17,20)(18,21)(19,22)(23,24)(25,27)(26,28);(5,6)(7,8)(13,16)(14,19)(15,21)(17,24)(18,22)(20,26)(23,25)(27,28);(1,2)(3,4)(5,6)(7,8)(13,17,18,27)(14,20,21,25)(15,23,19,26)(16,24,22,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=h,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df86 conjugacy classes — view the full character table.
| maximal subgroup of | #72266886 |
| maximal normal subgroup of | #72266886 |
| maximal quotient of | #72267036 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 991940);