Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393929 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
4338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(5,6)(7,8)(9,10,14,13)(11,15,19,18)(12,16,21,20)(17,23,24,22);(1,2)(3,4)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24);(2,4)(9,12)(10,16)(11,17)(13,20)(14,21)(15,22)(18,23)(19,24);(6,8)(9,13,14,10)(11,18,19,15)(12,20,21,16)(17,23,24,22)Polycyclic presentation:
2^9|a^2=i,d^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72249909, #72251632, #72251698, … |
| maximal normal subgroup of | #72249909, #72251632, #72251698, … |
| maximal quotient of | #72249963, #72257491, #72262662, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301128);