Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #498904 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2231 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(6,11)(7,8)(9,18)(12,19)(13,20)(14,16);(5,6)(7,11)(8,16)(9,12)(10,14)(13,18)(15,19)(17,20);(2,4)(6,12)(9,18)(11,19)(13,14)(15,17)(16,20);(5,7,10,8)(6,11,14,16)(9,15,18,17)(12,19,13,20)Polycyclic presentation:
2^9|d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,g^a=ghi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=eh,g^c=gi,g^e=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72231758, #72238397, #72238411, … |
| maximal normal subgroup of | #72231758, #72238397, #72238411, … |
| maximal quotient of | #72231752, #72231776, #72238399, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 406102);