Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #396106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2459 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14832 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
804 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9,4,17)(5,20)(6,14)(7,12)(8,13)(10,19,26,16)(11,21)(15,29,18,24)(22,28)(23,27)(25,30,32,31);(1,3,7,16,6,4,8,19)(2,9,12,10,14,17,13,26)(5,15,22,30,21,18,23,31)(11,24,27,25,20,29,28,32);(1,4,6,3)(2,10)(5,18,21,15)(7,19,8,16)(9,13)(11,25)(12,17)(14,26)(20,32)(22,31,23,30)(24,28)(27,29);(1,5,6,21)(2,11)(3,15,4,18)(7,22,8,23)(9,24)(10,25)(12,27)(13,28)(14,20)(16,30,19,31)(17,29)(26,32)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=e,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^b=gi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72279915 |
| maximal normal subgroup of | #72279915 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 309291);