Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #477178 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #17105 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #23444 | C2 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1276 conjugacy classes of subgroups; only the 168 normal subgroups are shown.
This group has 1276 conjugacy classes of subgroups — too many to draw. Only its 168 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)(3,7,13,19)(4,8,15,21)(5,9,16,23)(6,10)(11,17)(12,18)(14,20)(22,28)(24,29,26,32)(25,30)(27,31);(1,3)(2,7,10,19)(4,11,15,25)(5,12,16,27)(6,13)(8,17)(9,18)(14,26)(20,32,22,29)(21,30)(23,31)(24,28);(1,4)(2,8,10,21)(3,11,13,25)(5,14)(6,15)(7,17)(9,22,23,20)(12,26,27,24)(16,28)(18,29)(19,30)(31,32);(1,5)(2,9,10,23)(3,12,13,27)(4,14)(6,16)(7,18)(8,20,21,22)(11,24,25,26)(15,28)(17,29)(19,31)(30,32)Polycyclic presentation:
2^9|a^2=ef,b^2=f,c^2=e,d^2=e,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=def,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^c=gi,f^d=fi,h^d=hi47 conjugacy classes — view the full character table.
| maximal subgroup of | #72237732, #72237759, #72237763, … |
| maximal normal subgroup of | #72237732, #72237759, #72237763, … |
| maximal quotient of | #72265881 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 384374);