Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #251883 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2404 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | #29634 | C2 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1076 conjugacy classes of subgroups; only the 126 normal subgroups are shown.
This group has 1076 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,6)(4,7)(5,8)(9,11)(10,15)(12,14)(13,16)(17,18,21,24)(19,22,26,28)(20,23,27,30)(25,29,31,32);(1,2)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,19)(18,22)(20,25)(21,26)(23,29)(24,28)(27,31)(30,32);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(20,27)(22,28)(23,30)(25,31);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,20,21,27)(18,23,24,30)(19,25,26,31)(22,29,28,32)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^d=fh44 conjugacy classes — view the full character table.
| maximal subgroup of | #72240628, #72240635, #72251351, … |
| maximal normal subgroup of | #72240628, #72240635, #72251351, … |
| maximal quotient of | #72251470, #72251519 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 159095);