Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #436471 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15561 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1188 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,11,25,10)(3,13,16,19,8,18,15,4)(5,20,23,9,24,28,22,29)(14,17,21,31,26,32,30,27);(1,3,6,16,7,8,25,15)(2,4,12,13,11,19,10,18)(5,21,22,14,24,30,23,26)(9,27,20,32,29,31,28,17);(1,4,7,19)(2,8,11,3)(5,17,24,32)(6,18,25,13)(9,26,29,14)(10,15,12,16)(20,30,28,21)(22,31,23,27);(1,5)(2,9,11,29)(3,14,8,26)(4,17)(6,22)(7,24)(10,28,12,20)(13,27)(15,21,16,30)(18,31)(19,32)(23,25)Polycyclic presentation:
2^9|a^2=g,b^2=ef,c^2=i,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,g^c=gi,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72237070, #72237072, #72237084, … |
| maximal normal subgroup of | #72237070, #72237072, #72237084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 343668);