Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #453946 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16247 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
814 conjugacy classes of subgroups; only the 160 normal subgroups are shown.
This group has 814 conjugacy classes of subgroups — too many to draw. Only its 160 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,14,16,8)(5,17)(6,10)(9,19)(11,20,27,23)(12,21)(13,22)(15,28,30,24)(18,25)(26,32,29,31);(1,3)(2,7)(4,11,16,27)(5,18)(6,13)(8,23,14,20)(9,25)(10,22)(12,19)(15,29,30,26)(17,21)(24,31,28,32);(1,4)(2,8)(3,11,13,27)(5,15)(6,16)(7,20,22,23)(9,24)(10,14)(12,26,18,29)(17,28)(19,30)(21,31,25,32);(1,5)(2,9,10,17)(3,12,13,18)(4,15)(6,19)(7,21)(8,24,14,28)(11,26,27,29)(16,30)(20,31)(22,25)(23,32)Polycyclic presentation:
2^9|a^2=efg,b^2=efg,c^2=fgh,d^2=ef,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72318262, #72318356, #72318505, … |
| maximal normal subgroup of | #72318262, #72318356, #72318505, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361142);