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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2678 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3266 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,20)(17,18)(21,22,24,27)(23,26,28,25);(1,2,3,4)(6,9)(10,12)(13,15)(14,16)(17,20)(18,19)(21,23)(22,25)(24,28)(26,27);(1,2)(3,4)(7,11)(10,12)(14,18)(16,19)(21,23,24,28)(22,26,27,25);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,26)(24,27)(25,28);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,19)(18,20)(21,23)(22,26)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=gi,b^2=ghi,c^2=gi,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 | #72326818, #72326826, #72326907, … |
| maximal normal subgroup of | #72326818, #72326826, #72326907, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2188313);