Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437999 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
5066 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,9)(8,15,17,11)(10,12)(13,16,19,20)(14,18);(1,2)(3,4)(5,7)(6,9)(8,13)(10,14)(11,16)(12,18)(15,20)(17,19);(2,4)(5,8)(6,11)(7,13)(9,16)(10,17)(12,15)(14,19)(18,20);(2,4)(5,9)(6,7)(8,16)(10,18)(11,13)(12,14)(15,19)(17,20);(6,12)(9,18)(11,15)(16,20)Polycyclic presentation:
2^9|a^2=f,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,g^b=gi,f^c=fi,e^d=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72240655, #72240656, #72251367, … |
| maximal normal subgroup of | #72240655, #72240656, #72251367, … |
| maximal quotient of | #72238043, #72240675, #72267127, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345199);