Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #434718 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15506 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1114 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)(3,12,14,7)(4,16,18,8)(5,9)(6,10)(11,21)(13,27)(15,24,20,23)(17,22)(19,26,30,25)(28,32)(29,31);(1,3)(2,7)(4,17)(5,20)(6,14)(8,22)(9,24)(10,12)(11,23)(13,18)(15,21)(16,27)(19,28)(25,32)(26,31)(29,30);(1,4)(2,8)(3,13,14,17)(5,19)(6,18)(7,22,12,27)(9,25)(10,16)(11,26)(15,28,20,29)(21,30)(23,31,24,32);(2,9)(5,21)(7,23)(8,25)(10,11)(12,24)(15,20)(16,26)(19,30)(22,31)(27,32)(28,29)Polycyclic presentation:
2^9|a^2=fg,c^2=ef,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,f^b=fi,h^b=hi,g^c=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72338617, #72338620, #72338636 |
| maximal normal subgroup of | #72338617, #72338620, #72338636 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341914);