Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456381 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2712 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #16368 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
956 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8,11,21)(9,12,25,22)(10,26,28,23)(13,30)(17,31)(24,29,32,27);(1,2,5,21,7,11,6,8)(3,12,14,9,16,22,15,25)(4,10,20,26,19,28,18,23)(13,29,31,24,30,27,17,32);(1,3)(2,9)(4,17)(5,15)(6,14)(7,16)(8,22)(10,27)(11,25)(12,21)(13,18)(19,31)(20,30)(23,32)(24,26)(28,29);(1,4)(2,10)(3,13)(5,20)(6,18)(7,19)(8,23)(9,24)(11,28)(12,29)(14,31)(15,17)(16,30)(21,26)(22,27)(25,32)Polycyclic presentation:
2^9|a^2=eg,b^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72338916, #72338919, #72338926 |
| maximal normal subgroup of | #72338916, #72338919, #72338926 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 367802);