Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477203 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17105 |
| 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 | |
1119 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,14,8,11,7,3)(4,12,17,9,19,28,18,15)(5,20,21,13,22,10,23,26)(16,27,30,24,31,29,32,25);(1,3,6,11,8,14,7,2)(4,15,17,28,19,9,18,12)(5,10,21,13,22,20,23,26)(16,27,30,25,31,29,32,24);(1,4)(2,9,11,15)(3,12,14,28)(5,16)(6,17)(7,18)(8,19)(10,25,20,24)(13,29,26,27)(21,30)(22,31)(23,32);(1,5)(2,10,11,20)(3,13,14,26)(4,16)(6,21)(7,23)(8,22)(9,24,15,25)(12,27,28,29)(17,30)(18,32)(19,31)Polycyclic presentation:
2^9|a^2=ef,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,f^a=fi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,h^c=hi,g^d=gi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237063, #72237067, #72237090, … |
| maximal normal subgroup of | #72237063, #72237067, #72237090, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384399);