Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393929 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2104 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,17,20)(15,18,22,25)(16,19,23,26)(21,27,28,24);(1,2)(3,4)(13,15)(14,18)(16,21)(17,22)(19,24)(20,25)(23,28)(26,27);(2,4)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,28)(25,27);(5,7,8,12)(6,9,10,11)Polycyclic presentation:
2^9|a^2=hi,d^2=h,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 | #72249963, #72249970, #72249986, … |
| maximal normal subgroup of | #72249963, #72249970, #72249986, … |
| maximal quotient of | #72250465, #72250688, #72257195, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 321818);