Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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×#2410 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3925 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)(13,14,18,20)(15,17,16,19)(21,22)(23,27)(24,26)(25,28);(2,4)(6,9)(10,12)(13,15)(14,19)(16,18)(17,20)(21,22,24,26)(23,25,28,27);(1,2)(3,4)(7,11)(10,12)(14,20)(15,16)(23,28)(25,27);(5,7,8,11)(6,10,9,12)(13,16,18,15)(14,17,20,19)(21,23)(22,25)(24,28)(26,27);(5,6)(7,10)(8,9)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,25)(24,26)(27,28)Polycyclic presentation:
2^9|a^2=g,b^2=gi,d^2=fgi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
| maximal subgroup of | #72327686, #72327718, #72327882 |
| maximal normal subgroup of | #72327686, #72327718, #72327882 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1490093);