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 | C22×#121 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1492 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 32 points:
(5,6)(7,8)(9,10,12,13)(11,14,15,16)(17,18,24,30)(19,26,21,32)(20,27,31,25)(22,29,23,28);(1,2)(3,4)(10,13)(14,16)(17,19,22,31,24,21,23,20)(18,25,28,26,30,27,29,32);(2,4)(9,11)(10,14)(12,15)(13,16)(17,20,24,31)(18,26,30,32)(19,22,21,23)(25,29,27,28);(6,8)(11,15)(14,16)(17,21,22,20,24,19,23,31)(18,27,28,32,30,25,29,26)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=i,d^2=f,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265097, #72265136, #72265143, … |
| maximal normal subgroup of | #72265097, #72265136, #72265143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 366614);