Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34681 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2837 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(5,6)(7,8)(9,10,13,12,14,16,21,20)(11,15,19,18,17,22,24,23);(1,2)(3,4)(9,11)(10,15)(12,18)(13,19)(14,17)(16,22)(20,23)(21,24);(6,8)(11,17)(15,22)(18,23)(19,24);(2,4)(11,17)(15,22)(18,23)(19,24);(9,12,13,16,14,20,21,10)(11,18,19,22,17,23,24,15)Polycyclic presentation:
2^9|a^2=h,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=dfi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72257505, #72257523, #72257677, … |
| maximal normal subgroup of | #72257505, #72257523, #72257677, … |
| maximal quotient of | #72307967 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291218);