Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454375 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1300 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,3,4)(5,6)(7,8)(10,12)(14,15)(17,18,21,24,22,25,20,30)(19,23,28,31,29,32,27,26);(9,10,13,12)(11,15,16,14)(17,19,21,28,22,29,20,27)(18,23,24,31,25,32,30,26);(2,4)(9,11,13,16)(10,14,12,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,27)(28,32)(29,31);(6,8)(9,12,13,10)(11,14,16,15)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,31,32,26)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=e,d^2=ei,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72264389, #72264400, #72264430, … |
| maximal normal subgroup of | #72264389, #72264400, #72264430, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372046);