Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #501096 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×D4 | C24 |
| LCS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,11)(8,13,18,12)(9,14)(10,16,20,17)(15,19);(1,2)(3,4)(5,7,9,15)(6,10,14,20)(8,17,18,16)(11,13,19,12);(2,4)(5,7)(6,11)(8,17)(9,15)(10,12)(13,20)(14,19)(16,18);(6,12)(7,15)(8,18)(10,19)(11,20)(13,14)Polycyclic presentation:
2^9|a^2=f,b^2=i,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fhi,d^c=di,e^c=eh,e^d=ei,f^d=fhi,f^e=fi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72250097, #72256807, #72256811, … |
| maximal normal subgroup of | #72250097, #72256807, #72256811, … |
| maximal quotient of | #72250195 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 408299);