Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #451047 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15984 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3068 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1278 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,22,7,11,20,10)(3,12,21,24,14,29,5,8)(4,9,17,27,19,28,18,26)(13,25,15,23,31,30,16,32);(1,3)(2,8)(4,15)(5,20)(6,21)(7,14)(9,25)(10,29)(11,24)(12,22)(13,18)(16,19)(17,31)(23,27)(26,32)(28,30);(1,4)(2,9)(3,13)(5,16)(6,17)(7,19)(8,23)(10,26)(11,28)(12,30)(14,31)(15,21)(18,20)(22,27)(24,32)(25,29);(1,5,20,3,7,21,6,14)(2,8,22,29,11,24,10,12)(4,16,18,13,19,15,17,31)(9,23,27,25,28,32,26,30)Polycyclic presentation:
2^9|a^2=gh,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338625, #72338626, #72338632 |
| maximal normal subgroup of | #72338625, #72338626, #72338632 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 358244);