Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #439368 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15635 |
| 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 | |
1168 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,23,7,10,12,11)(3,8,4,16,14,25,15,26)(5,18,20,27,22,28,21,9)(13,29,17,31,19,32,30,24);(1,3,12,4,7,14,6,15)(2,8,23,26,10,25,11,16)(5,19,20,17,22,13,21,30)(9,24,28,29,27,31,18,32);(1,4,6,14,7,15,12,3)(2,8,23,16,10,25,11,26)(5,17,20,19,22,30,21,13)(9,24,18,29,27,31,28,32);(1,5)(2,9)(3,13)(4,17)(6,20)(7,22)(8,24)(10,27)(11,28)(12,21)(14,19)(15,30)(16,29)(18,23)(25,31)(26,32)Polycyclic presentation:
2^9|a^2=ef,b^2=efg,c^2=ef,f^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^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338910, #72338919, #72338920, … |
| maximal normal subgroup of | #72338910, #72338919, #72338920, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 346564);