Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #464970 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16512 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23407 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1148 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,13)(4,18)(5,6)(9,23)(11,12)(16,17)(19,29)(20,21)(22,27)(24,32)(26,31);(1,2,6,11,7,12,5,8)(3,14,13,23,17,25,16,9)(4,10,20,27,21,28,18,22)(15,26,19,31,30,32,29,24);(1,3,5,13,7,17,6,16)(2,9,8,14,12,23,11,25)(4,19,18,30,21,29,20,15)(10,26,22,31,28,32,27,24);(1,4)(2,10)(3,15)(5,18)(6,20)(7,21)(8,22)(9,24)(11,27)(12,28)(13,19)(14,26)(16,29)(17,30)(23,31)(25,32)Polycyclic presentation:
2^9|b^2=g,c^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,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72318217, #72318883, #72318969, … |
| maximal normal subgroup of | #72318217, #72318883, #72318969, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372166);