Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4311730 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2404 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3436 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)(6,9)(10,12)(13,14,17,16)(15,19,20,18)(22,25)(23,27);(14,16)(18,19)(21,22)(23,28)(24,25)(26,27);(1,2,3,4)(15,20)(18,19)(22,25)(26,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,19)(17,20)(21,22)(23,26)(24,25)(27,28);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(21,23,24,27)(22,26,25,28)Polycyclic presentation:
2^9|a^2=fh,c^2=g,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72334099, #72334102, #72334156 |
| maximal normal subgroup of | #72334099, #72334102, #72334156 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4270167);