Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #35680 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C23×H(5,𝔽2) | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
19886 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(6,8)(13,14)(15,18)(16,19)(17,20);(10,12)(13,15)(14,17)(16,20)(18,19);(1,2,3,4)(5,6,7,8)(9,10,11,12);(13,15)(14,18)(16,20)(17,19);(13,14)(15,17)(16,19)(18,20)Polycyclic presentation:
2^9|c^2=i,b^a=bf,c^a=cg,c^b=ch,e^d=ef128 conjugacy classes — view the full character table.
| maximal subgroup of | #72214239, #72214271, #72214449, … |
| maximal normal subgroup of | #72214239, #72214271, #72214449, … |
| maximal quotient of | C2×#517592, #72250140, #72250146, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 424788);