Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
9023 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20);(6,9)(10,12)(13,15)(14,17)(16,20)(18,19);(2,3)(13,14)(15,19)(16,18)(17,20);(1,3,4,2)(15,20)(17,19);(14,18)(17,19);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^9|a^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,d^c=dh,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72234450, #72234451, #72234464, … |
| maximal normal subgroup of | #72234450, #72234451, #72234464, … |
| maximal quotient of | #72251184, #72251272, #72256899, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7574582);