Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #245225 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
1803 conjugacy classes of subgroups; only the 156 normal subgroups are shown.
This group has 1803 conjugacy classes of subgroups — too many to draw. Only its 156 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:
(1,2,5,10)(3,12,13,26)(4,8,17,24)(6,11,18,9)(7,19,21,14)(15,28,29,27)(16,23,31,25)(20,30,22,32);(4,16)(7,20)(8,23)(12,26)(14,30)(17,31)(19,32)(21,22)(24,25)(27,28);(1,3,6,15)(2,7,11,22)(4,14,16,30)(5,13,18,29)(8,12,23,28)(9,20,10,21)(17,19,31,32)(24,26,25,27);(4,17)(8,24)(12,26)(14,19)(16,31)(23,25)(27,28)(30,32)Polycyclic presentation:
2^9|a^2=h,c^2=i,b^a=be,c^a=cf,d^a=de,g^a=ghi,c^b=cg,f^b=fi,e^c=eh,f^d=fh56 conjugacy classes — view the full character table.
| maximal subgroup of | #72234113, #72241248, #72247257, … |
| maximal normal subgroup of | #72234113, #72241248, #72247257, … |
| maximal quotient of | #72209867, #72241253, #72247316, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 152432);